Search: a001481 -id:a001481
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A372917
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a(n) is the number of distinct rectangles with area n whose vertices lie on points of a unit square grid.
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1
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1, 2, 1, 3, 2, 3, 1, 4, 2, 5, 1, 5, 2, 3, 3, 5, 2, 5, 1, 8, 2, 3, 1, 7, 3, 5, 2, 5, 2, 9, 1, 6, 2, 5, 3, 8, 2, 3, 3, 11, 2, 6, 1, 5, 5, 3, 1, 9, 2, 8, 3, 8, 2, 6, 3, 7, 2, 5, 1, 15, 2, 3, 3, 7, 5, 6, 1, 8, 2, 9, 1, 11, 2, 5, 5, 5, 2, 9, 1, 14, 3, 5, 1, 10, 5, 3
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OFFSET
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1,2
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COMMENTS
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A rectangle in the square unit grid has the sides W = w*sqrt(r) and H = h*sqrt(r). The area is therefore n = w*h*r. Let r be a squarefree divisor of n that can be written as the sum of two squares x^2 + y^2. The number of distinct rectangles is then the sum of the number of ways for each value of r to decompose n/r into two factors w and h (with w >= h).
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LINKS
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FORMULA
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a(n) = ceiling(Product_{i=1..omega(n)}(k[i]*e[i] + 1)/2), with k[i] = 2 if p[i] mod 4 = 3 and k[i] = 1 else, where p[i]^e[i] is the prime factorization of n.
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EXAMPLE
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See also the linked illustrations of the terms a(4) = 3, a(8) = 4, a(15) = 3.
n = 4 has the three divisors 1, 2, 4. Since 4 is not squarefree, r can have the values 1 or 2. For r = 1 = 1^2 + 0^2 there are two rectangles (2,2), (4,1). For r = 2 = 1^2 + 1^2 and n/r = 4/2 = 2 = w*h there is the rectangle (2*sqrt(2), 1*sqrt(2)). That's a total of a(4) = 3 distinct rectangles.
n = 8 has the four divisors 1, 2, 4, 8. Since 4 and 8 are not squarefree, r can have the values 1 or 2. For r = 1 = 1^2 + 0^2 there are two rectangles (4,2), (8,1). For r = 2 = 1^2 + 1^2 and n/r = 8/2 = 4 = w*h there are the rectangles (4*sqrt(2), 1*sqrt(2)) and (2*sqrt(2), 2*sqrt(2)). That's a total of a(8) = 4 distinct rectangles.
n = 15 has the four divisors 1, 3, 5, 15. They are all squarefree, but 3 and 15 cannot be written as a sum of two squares, r can only have the values 1 or 5. For r = 1 = 1^2 + 0^2 there are two rectangles (5,3), (15,1). For r = 5 = 2^2 + 1^2 and n/r = 15/5 = 3 = w*h there is the rectangles (3*sqrt(5), 1*sqrt(5)). That's a total of a(15) = 3 distinct rectangles.
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MAPLE
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local f, i, prod;
f:=ifactors(n)[2];
prod:=1;
for i from 1 to numelems(f) do
if f[i][1] mod 4 = 3 then
prod:=prod*(1*f[i][2]+1);
else
prod:=prod*(2*f[i][2]+1);
end if;
end do;
return round(prod/2);
end proc;
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PROG
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(PARI) a(n) = my(f=factor(n)); prod(i=1, #f[, 1], if(f[i, 1]%4==3, 1, 2)*f[i, 2] + 1) \/ 2; \\ Kevin Ryde, Jun 09 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A111908
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Numbers that are not the sum of a prime and a nonzero triangular number.
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1
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1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886
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OFFSET
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1,2
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COMMENTS
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Can anybody prove or disprove a(n) = O(n^c) for some constant c?
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LINKS
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EXAMPLE
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7 = 1+6 = 2+5 = 3+4; 7 is in the sequence because there is no sum where the first element is a prime and the second one a triangular number.
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MATHEMATICA
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lim=6000; plim=PrimePi[lim]; tlim=Ceiling[Sqrt[2lim]]; Complement[Range[lim], Union[Flatten[Table[Prime[i]+PolygonalNumber[j], {i, plim}, {j, tlim}]]]] (* James C. McMahon, Jun 04 2024 *)
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CROSSREFS
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Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A002654
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Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).
(Formerly M0012 N0001)
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105
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1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2, 0, 0, 2, 0
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OFFSET
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1,5
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COMMENTS
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Number of sublattices of Z X Z of index n that are similar to Z X Z; number of (principal) ideals of Z[i] of norm n.
a(n) is also one fourth of the number of integer solutions of n = x^2 + y^2 (order and signs matter, and 0 (without signs) is allowed). a(n) = N(n)/4, with N(n) from p. 147 of the Niven-Zuckermann reference. See also Theorem 5.12, p. 150, which defines a (strongly) multiplicative function h(n) which coincides with A056594(n-1), n >= 1, and N(n)/4 = sum(h(d), d divides n). - Wolfdieter Lang, Apr 19 2013
a(2+8*N) = A008441(N) gives the number of ways of writing N as the sum of 2 (nonnegative) triangular numbers for N >= 0. - Wolfdieter Lang, Jan 12 2017
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -4. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022
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REFERENCES
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J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.
George Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed., Chelsea Publishing Co., New York, 1959, Part II, p. 346 Exercise XXI(17). MR0121327 (22 #12066)
Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley, 1980, pp. 147 and 150.
Günter Scheja and Uwe Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 340.
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LINKS
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Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., The Mathematics of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167. [Annotated scanned copy]
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FORMULA
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Dirichlet series: (1-2^(-s))^(-1)*Product (1-p^(-s))^(-2) (p=1 mod 4) * Product (1-p^(-2s))^(-1) (p=3 mod 4) = Dedekind zeta-function of Z[ i ].
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16.
If n=2^k*u*v, where u is product of primes 4m+1, v is product of primes 4m+3, then a(n)=0 unless v is a square, in which case a(n) = number of divisors of u (Jacobi).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p == 1 (mod 4); (e+1) mod 2 if p == 3 (mod 4). - David W. Wilson, Sep 01 2001
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * (4*w + 1). - Michael Somos, Jul 19 2004
G.f.: Sum_{n>=1} ((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n)). - Vladeta Jovovic, Sep 15 2004
Expansion of (eta(q^2)^10 / (eta(q) * eta(q^4))^4 - 1)/4 in powers of q.
G.f.: Sum_{k>0} x^k / (1 + x^(2*k)) = Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 - x^(2*k - 1)). - Michael Somos, Aug 17 2005
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(2*n) = a(n). - Michael Somos, Nov 01 2006
a(n) = Sum_{d|n} A056594(d-1), n >= 1. See the above comment on A056594(d-1) = h(d) of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
Dirichlet g.f.: zeta(s)*beta(s) = zeta(s)*L(chi_2(4),s). - Ralf Stephan, Mar 27 2015
G.f.: (theta_3(x)^2 - 1)/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
Sum_{n>=1} (-1)^n*a(n)/n = Pi*log(2)/4 (Covo, 2010). - Amiram Eldar, Apr 07 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 = 0.785398... (A003881). - Amiram Eldar, Oct 11 2022
Sum_{k=1..n} a(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =
4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...
The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576... is wrong! (End)
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EXAMPLE
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4 = 2^2, so a(4) = 1; 5 = 1^2 + 2^2 = 2^2 + 1^2, so a(5) = 2.
x + x^2 + x^4 + 2*x^5 + x^8 + x^9 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + x^18 + ...
2 = (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2 = (-1)^2 + (+1)^2 = (-1)^2 + (-1)^2. Hence there are 4 integer solutions, called N(2) in the Niven-Zuckerman reference, and a(2) = N(2)/4 = 1. 4 = 0^1 + (+2)^2 = (+2)^2 + 0^2 = 0^2 + (-2)^2 = (-2)^2 + 0^2. Hence N(4) = 4 and a(4) = N(4)/4 = 1. N(5) = 8, a(5) = 2. - Wolfdieter Lang, Apr 19 2013
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MAPLE
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with(numtheory):
local count1, count3, d;
count1 := 0:
count3 := 0:
for d in numtheory[divisors](n) do
if d mod 4 = 1 then
count1 := count1+1
elif d mod 4 = 3 then
count3 := count3+1
fi:
end do:
count1-count3;
end proc:
# second Maple program:
a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
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MATHEMATICA
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a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1] - Count[Divisors[n], d_ /; Mod[d, 4] == 3]; a/@Range[105] (* Jean-François Alcover, Apr 06 2011, after R. J. Mathar *)
QP = QPochhammer; CoefficientList[(1/q)*(QP[q^2]^10/(QP[q]*QP[q^4])^4-1)/4 + O[q]^100, q] (* Jean-François Alcover, Nov 24 2015 *)
f[2, e_] := 1; f[p_, e_] := If[Mod[p, 4] == 1, e + 1, Mod[e + 1, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)
Rest[CoefficientList[Series[EllipticTheta[3, 0, q]^2/4, {q, 0, 100}], q]] (* Vaclav Kotesovec, Mar 10 2023 *)
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PROG
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(PARI) direuler(p=2, 101, 1/(1-X)/(1-kronecker(-4, p)*X))
(PARI) {a(n) = polcoeff( sum(k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}
(PARI) {a(n) = sumdiv( n, d, (d%4==1) - (d%4==3))}
(PARI) {a(n) = local(A); A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / (eta(x + A) * eta(x^4 + A))^4 / 4, n)} \\ Michael Somos, Jun 03 2005
(PARI) a(n)=my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, (f[i, 2]+1)%2)) \\ Charles R Greathouse IV, Sep 09 2014
(PARI) my(B=bnfinit(x^2+1)); vector(100, n, #bnfisintnorm(B, n)) \\ Joerg Arndt, Jun 01 2024
(Haskell)
a002654 n = product $ zipWith f (a027748_row m) (a124010_row m) where
f p e | p `mod` 4 == 1 = e + 1
| otherwise = (e + 1) `mod` 2
m = a000265 n
(Python)
from math import prod
from sympy import factorint
def A002654(n): return prod(1 if p == 2 else (e+1 if p % 4 == 1 else (e+1) % 2) for p, e in factorint(n).items()) # Chai Wah Wu, May 09 2022
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CROSSREFS
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Cf. A002175, A008441, A121444, A122856, A122865, A022544, A143574, A000265, A027748, A124010, A025426 (two squares, order does not matter), A120630 (Dirichlet inverse), A101455 (Mobius transform), A000089, A241011.
If one simply reads the table in Glaisher, PLMS 1884, which omits the zero entries, one gets A213408.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
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KEYWORD
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core,easy,nonn,nice,mult
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AUTHOR
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STATUS
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approved
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A064533
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Decimal expansion of Landau-Ramanujan constant.
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41
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7, 6, 4, 2, 2, 3, 6, 5, 3, 5, 8, 9, 2, 2, 0, 6, 6, 2, 9, 9, 0, 6, 9, 8, 7, 3, 1, 2, 5, 0, 0, 9, 2, 3, 2, 8, 1, 1, 6, 7, 9, 0, 5, 4, 1, 3, 9, 3, 4, 0, 9, 5, 1, 4, 7, 2, 1, 6, 8, 6, 6, 7, 3, 7, 4, 9, 6, 1, 4, 6, 4, 1, 6, 5, 8, 7, 3, 2, 8, 5, 8, 8, 3, 8, 4, 0, 1, 5, 0, 5, 0, 1, 3, 1, 3, 1, 2, 3, 3, 7, 2, 1, 9, 3, 7, 2, 6, 9, 1, 2, 0, 7, 9, 2, 5, 9, 2, 6, 3, 4, 1, 8, 7, 4, 2, 0, 6, 4, 6, 7, 8, 0, 8, 4, 3, 2, 3, 0, 6, 3, 3, 1, 5, 4, 3, 4, 6, 2, 9, 3, 8, 0, 5, 3, 1, 6, 0, 5, 1, 7, 1, 1, 6, 9, 6, 3, 6, 1, 7, 7, 5, 0, 8, 8, 1, 9, 9, 6, 1, 2, 4, 3, 8, 2, 4, 9, 9, 4, 2, 7, 7, 6, 8, 3, 4, 6, 9, 0, 5, 1, 6, 2, 3, 5, 1, 3, 9, 2, 1, 8, 7, 1, 9, 6, 2, 0, 5, 6, 9, 0, 5, 3, 2, 9, 5, 6, 4, 4, 6, 7, 0, 4
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OFFSET
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0,1
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COMMENTS
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Named after the German mathematician Edmund Georg Hermann Landau (1877-1938) and the Indian mathematician Srinivasa Ramanujan (1887-1920). - Amiram Eldar, Jun 20 2021
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REFERENCES
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Bruce C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, pp. 52, 60-66; MR 95e: 11028.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
G. H. Hardy, "Ramanujan, Twelve lectures on subjects suggested by his life and work", Chelsea, 1940, pp. 60-63; MR 21 # 4881.
Edmund Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys., 13, 1908, pp. 305-312.
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LINKS
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FORMULA
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Equals (Pi/4) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2)^(1/2).
Equals (1/sqrt(2)) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2)^(-1/2).
Equals (1/sqrt(2)) * Product_{k>=1} ((1 - 1/2^(2^k)) * zeta(2^k)/beta(2^k)), where beta is the Dirichlet beta function (Shanks, 1964). (End)
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EXAMPLE
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0.76422365358922066299069873125009232811679054139340951472168667374...
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MATHEMATICA
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First@ RealDigits@ N[1/Sqrt@2 Product[((1 - 2^(-2^k)) 4^(2^k) Zeta[2^k]/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^(2^(-k - 1)), {k, 8}], 2^8] (* Robert G. Wilson v, Jul 01 2007 *)
(* Victor Adamchik calculated 5100 digits of the Landau-Ramanujan constant using Mathematica (from Mathematica 4 demos): *) LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}]/Sqrt[2], n]];
(* The code reported here is the code at https://library.wolfram.com/infocenter/Demos/120/. Looking carefully at the outputs reported there one sees that: the last 8 digits of the 500-digit output ("74259724") are not the same as those listed in the 1000-digit output ("94247095"); the same happens with the last 18 digits of the 1000-digit output ("584868265713856413") and the corresponding ones in the 5100-digit output ("852514327407923660"). - Alessandro Languasco, May 07 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More references needed! Hardy and Wright? Gruber and Lekkerkerker?
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STATUS
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approved
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A069011
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Triangle with T(n,k) = n^2 + k^2.
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+0
7
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0, 1, 2, 4, 5, 8, 9, 10, 13, 18, 16, 17, 20, 25, 32, 25, 26, 29, 34, 41, 50, 36, 37, 40, 45, 52, 61, 72, 49, 50, 53, 58, 65, 74, 85, 98, 64, 65, 68, 73, 80, 89, 100, 113, 128, 81, 82, 85, 90, 97, 106, 117, 130, 145, 162, 100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200
(list;
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internal format)
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OFFSET
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0,3
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COMMENTS
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For any i,j >=0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013
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LINKS
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FORMULA
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T(n+1,k+1) = T(n,k) + 2*(n+k+1), k=0..n; T(n+1,0) = T(n,0) + 2*n + 1. - Reinhard Zumkeller, Oct 11 2013
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EXAMPLE
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Triangle T(n,k) begins:
0;
1, 2;
4, 5, 8;
9, 10, 13, 18;
16, 17, 20, 25, 32;
25, 26, 29, 34, 41, 50;
36, 37, 40, 45, 52, 61, 72;
49, 50, 53, 58, 65, 74, 85, 98;
64, 65, 68, 73, 80, 89, 100, 113, 128;
81, 82, 85, 90, 97, 106, 117, 130, 145, 162;
100, 101, 104, 109, 116, 125, 136, 149, 164, 181, 200;
...
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PROG
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(Haskell)
a069011 n k = a069011_tabl !! n !! k
a069011_row n = a069011_tabl !! n
a069011_tabl = map snd $ iterate f (1, [0]) where
f (i, xs@(x:_)) = (i + 2, (x + i) : zipWith (+) xs [i + 1, i + 3 ..])
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CROSSREFS
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Cf. A001481 for terms in this sequence, A000161 for number of times each term appears, A048147 for square array.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A372394
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Determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*n+1)]_{1<i,j<2*n}, where Jacobi(a,m) denotes the Jacobi symbol (a/m).
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1
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0, 0, 0, 33, 0, 0, 0, -77539, 1811939328, -405798912, 0, 0, 649564705105200, -2787119627540625, 86463597248512, 0, 0, 0, 353143905335474188320, -66016543975248459410178048, 0, 23092056382629010556862857216, 0, 0, 0, 0, -5310136941067623723354761986048
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OFFSET
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2,4
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COMMENTS
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Conjecture: (i) If n == 6, 8 (mod 10), and 2*n + 1 is a sum of two squares, then a(n) = 0.
(ii) If n == 5, 9 (mod 10), then a(n) is not relatively prime to 2*n + 1.
See also A372314 for other similar conjectures.
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LINKS
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EXAMPLE
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a(2) = 0 since the determinant of the matrix [Jacobi(i^2+5*i*j+5*j^2,2*2+1)]_{1<i,j<2*2} = [1,1;1,1] has the value 0.
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MATHEMATICA
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a[n_]:=a[n]=Det[Table[JacobiSymbol[i^2+5*i*j+5*j^2, 2n+1], {i, 2, 2n-1}, {j, 2, 2n-1}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 28}]; Print[tab]
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PROG
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(Python)
from sympy import Matrix, jacobi_symbol
def A372394(n): return Matrix(n-1<<1, n-1<<1, [jacobi_symbol(i*(i+5*j+14)+j*(5*j+30)+44, (n<<1)|1) for i in range(n-1<<1) for j in range(n-1<<1)]).det() # Chai Wah Wu, Apr 30 2024
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A372314
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Determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 2*n + 1)]_{1 < i, j < 2*n}, where Jacobi(a, m) denotes the Jacobi symbol (a / m).
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+0
3
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1, 3, 0, 125, -1215, 0, 0, 9126441, 0, -187590821, 0, 0, 20686753425, 0, 0, 0, 9224101117395305225, 0, 881852208012283730302080, 624391710361368134976, 0, -3428714319207136609529065, 0, 0, 3878246452353765171209988566241, 0, 0, 4308304210666498856284267223158421
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OFFSET
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2,2
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COMMENTS
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Conjecture 1: Let n be any positive integer.
(i) If a(2*n) is nonzero, then 4*n + 1 is a sum of two squares.
(ii) a(2*n + 1) is divisible by phi(4*n + 3)/2, where phi is Euler's totient function. If n is even, then a(2*n + 1)/(phi(4*n + 3)/2) is a square. This has been verified for n = 2..1000.
For any odd integer n > 3 and integers c and d, we introduce the notation: {c,d}_n = det[Jacobi(i^2 + c*i*j + d*j^2, n)]_{1 < i, j < n-1}.
The following conjecture is similar to Conjecture 1.
Conjecture 2: (1) {2, 2}_p = 0 for any prime p == 13,19 (mod 24), and {2, 2}_p == 0 (mod p) for any prime p == 17,23 (mod 24).
(2) If n == 5 (mod 8), then {4, 2}_n = 0. If n == 5 (mod 12), then {3, 3}_n = 0.
(3) If n == 5 (mod 12) and n is a sum of two squares, then {10, 9}_n = 0. Also, {10, 9}_p == 0 (mod p) for any prime p == 11 (mod 12).
(4) {8, 18}_p == 0 (mod p^2) for any prime p == 19 (mod 24), and {8,18}_p == 0 (mod p) for any prime p == 23 (mod 24). If n == 13,17 (mod 24) and n is a sum of two squares, then {8, 18}_n = 0.
We have verified Conjecture 2 for p or n smaller than 2000.
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LINKS
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EXAMPLE
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a(2) = 1 since the determinant of the matrix [Jacobi(i^2 + 3*i*j + 2*j^2, 5)]_{1 < i, j < 2*2} = [1,0; 0,1] is 1.
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MATHEMATICA
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a[n_]:=a[n]=Det[Table[JacobiSymbol[i^2+3*i*j+2*j^2, 2n+1], {i, 2, 2n-1}, {j, 2, 2n-1}]];
tab={}; Do[tab=Append[tab, a[n]], {n, 2, 29}]; Print[tab]
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PROG
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(PARI) f(i, j) = i^2 + 3*i*j + 2*j^2;
a(n) = matdet(matrix(2*n-2, 2*n-2, i, j, kronecker(f(i+1, j+1), 2*n+1)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A000404
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Numbers that are the sum of 2 nonzero squares.
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+0
235
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2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 169, 170, 173, 178
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OFFSET
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1,1
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COMMENTS
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From the formula it is easy to see that if k is in this sequence, then so are all odd powers of k. - T. D. Noe, Jan 13 2009
Also numbers whose cubes are the sum of two nonzero squares. - Joe Namnath and Lawrence Sze
A line perpendicular to y=mx has its first integral y-intercept at a^2+b^2. The remaining ones for that slope are multiples of that primitive value. - Larry J Zimmermann, Aug 19 2010
The primes in this sequence are sequence A002313.
If the two squares are not equal, then any power is still in the sequence: if k = x^2 + y^2 with x != y, then k^2 = (x^2-y^2)^2 + (2xy)^2 and k^3 = (x(x^2-3y^2))^2 + (y(3x^2-y^2))^2, etc. - Carmine Suriano, Jul 13 2012
There are never more than 3 consecutive terms that differ by 1. Triples of consecutive terms that differ by 1 occur infinitely many times, for example, 2(k^2 + k)^2, (k^2 - 1)^2 + (k^2 + 2 k)^2, and (k^2 + k - 1)^2 + (k^2 + k + 1)^2 for any integer k > 1. - Ivan Neretin, Mar 16 2017 [Corrected by Jerzy R Borysowicz, Apr 14 2017]
Number of terms less than 10^k, k=1,2,3,...: 3, 34, 308, 2690, 23873, 215907, 1984228, ... - Muniru A Asiru, Feb 01 2018
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REFERENCES
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David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 103.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 75, Theorem 4, with Theorem 2, p. 15.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
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LINKS
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FORMULA
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Let k = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s. Then k is a term iff 1) b_j == 0 (mod 2) for j=1..s and 2) r > 0 or t == 1 (mod 2) (or both).
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533.
There are B(x) = (x/sqrt(log x)) * (K + B2/log x + O(1/log^2 x)) terms of this sequence up to x, where K = A064533 and B2 = A227158. (End)
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EXAMPLE
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25 = 3^2 + 4^2, therefore 25 is a term. Note that also 25^3 = 15625 = 44^2 + 117^2, therefore 15625 is a term.
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MAPLE
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nMax:=178: A:={}: for i to floor(sqrt(nMax)) do for j to floor(sqrt(nMax)) do if i^2+j^2 <= nMax then A := `union`(A, {i^2+j^2}) else end if end do end do: A; # Emeric Deutsch, Jan 02 2017
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MATHEMATICA
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nMax=1000; n2=Floor[Sqrt[nMax-1]]; Union[Flatten[Table[a^2+b^2, {a, n2}, {b, a, Floor[Sqrt[nMax-a^2]]}]]]
Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Mar 24 2016 *)
Module[{upto=200}, Select[Union[Total/@Tuples[Range[Sqrt[upto]]^2, 2]], #<= upto&]] (* Harvey P. Dale, Sep 18 2021 *)
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PROG
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(PARI) is_A000404(n)= for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)) \\ M. F. Hasler, Feb 07 2009
(PARI) list(lim)=my(v=List(), x2); lim\=1; for(x=1, sqrtint(lim-1), x2=x^2; for(y=1, sqrtint(lim-x2), listput(v, x2+y^2))); Set(v) \\ Charles R Greathouse IV, Apr 30 2016
(Haskell)
import Data.List (findIndices)
a000404 n = a000404_list !! (n-1)
a000404_list = findIndices (> 0) a025426_list
(Magma) lst:=[]; for n in [1..178] do f:=Factorization(n); if IsSquare(n) then for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 1 then Append(~lst, n); break; end if; end for; else t:=0; for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 3 and d[2] mod 2 eq 1 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, n); end if; end if; end for; lst; // Arkadiusz Wesolowski, Feb 16 2017
(GAP) P:=List([1..10^4], i->i^2);;
(Python)
from itertools import count, islice
from sympy import factorint
def A000404_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue, 1)):
c = False
for p in (f:=factorint(n)):
if (q:= p & 3)==3 and f[p]&1:
break
elif q == 1:
c = True
else:
if c or f.get(2, 0)&1:
yield n
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CROSSREFS
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A001481 gives another version (allowing for zero squares).
Cf. A004431 (2 distinct squares), A063725 (number of representations), A024509 (numbers with multiplicity), A025284, A018825. Also A050803, A050801, A001105, A033431, A084888, A000578, A000290, A057961, A232499, A007692.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Erroneous Mathematica program fixed by T. D. Noe, Aug 07 2009
PARI code fixed for versions > 2.5 by M. F. Hasler, Jan 01 2013
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STATUS
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approved
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A371014
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The number of divisors of n that are the sum of 2 squares.
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+0
2
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1, 2, 1, 3, 2, 2, 1, 4, 2, 4, 1, 3, 2, 2, 2, 5, 2, 4, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 2, 4, 1, 6, 1, 4, 2, 6, 2, 2, 2, 8, 2, 2, 1, 3, 4, 2, 1, 5, 2, 6, 2, 6, 2, 4, 2, 4, 1, 4, 1, 6, 2, 2, 2, 7, 4, 2, 1, 6, 1, 4, 1, 8, 2, 4, 3, 3, 1, 4, 1, 10, 3, 4, 1, 3, 4, 2, 2
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = floor(e/2) + 1 if p == 3 (mod 4), and e+1 otherwise.
a(n) = 1 if and only if n is in A167181.
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MATHEMATICA
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f[p_, e_] := If[Mod[p, 4] == 3, Floor[e/2] + 1, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%4 == 3, f[i, 2]\2 + 1, f[i, 2] + 1)); }
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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A371015
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The largest divisor of n that is the sum of 2 squares.
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+0
2
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1, 2, 1, 4, 5, 2, 1, 8, 9, 10, 1, 4, 13, 2, 5, 16, 17, 18, 1, 20, 1, 2, 1, 8, 25, 26, 9, 4, 29, 10, 1, 32, 1, 34, 5, 36, 37, 2, 13, 40, 41, 2, 1, 4, 45, 2, 1, 16, 49, 50, 17, 52, 53, 18, 5, 8, 1, 58, 1, 20, 61, 2, 9, 64, 65, 2, 1, 68, 1, 10, 1, 72, 73, 74, 25
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(2*floor(e/2)) if p == 3 (mod 4), and p^e otherwise.
a(n) = n if and only if n is in A001481.
a(n) = 1 if and only if n is in A167181.
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MATHEMATICA
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f[p_, e_] := If[Mod[p, 4] == 3, p^(2*Floor[e/2]), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^if(f[i, 1]%4 == 3, 2*(f[i, 2]\2), f[i, 2])); }
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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