Search: a088534 -id:a088534
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0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 931, 1183, 1519, 1729, 2821, 3367
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OFFSET
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1,3
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COMMENTS
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Least squared radius of a circle around a grid point of the hexagonal lattice such that the number of grid points on this circle is not smaller than the number of grid points on any circle around a grid point of the hexagonal lattice with smaller radius. a(1) = 0 by convention.
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LINKS
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PROG
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(PARI) \\ after M. F. Hasler's A088534(n, d)
r2=0; for(k=0, 3500, my(a088534=sum(x=0, sqrt(k\3), sum(y=max(x, sqrtint(k-x^2)\2), sqrtint(k-2*x^2), x^2+x*y+y^2==k))); if(a088534>=r2, r2=a088534; print1(k, ", ")))
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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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A003627
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Primes of the form 3n-1.
(Formerly M1388)
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+10
130
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2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587
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OFFSET
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1,1
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COMMENTS
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Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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MAPLE
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t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1), 3*n+2}; fi; od: A003627 := convert(t1, list);
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MATHEMATICA
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Select[Prime[Range[200]], Mod[#, 3]==2&] (* Harvey P. Dale, Jan 31 2023 *)
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PROG
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(Magma) [n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011
(Haskell)
a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A003136
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Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice.
(Formerly M2336)
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+10
120
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0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192
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OFFSET
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1,3
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COMMENTS
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Equally, numbers of the form x^2 - xy + y^2. - Ray Chandler, Jan 27 2009
Also, numbers of the form X^2+3Y^2 (X=y+x/2, Y=x/2), cf. A092572. - Zak Seidov, Jan 20 2009
Theorem (Schering, Delone, Watson): The only positive definite binary quadratic forms that represent the same numbers are x^2+xy+y^2 and x^2+3y^2 (up to scaling). - N. J. A. Sloane, Jun 22 2014
Equivalently, numbers n such that the coefficient of x^n in Theta3(x)*Theta3(x^3) is nonzero. - Joerg Arndt, Jan 16 2011
Equivalently, numbers n such that the coefficient of x^n in a(x) (resp. b(x)) is nonzero where a(), b() are cubic AGM functions. - Michael Somos, Jan 16 2011
Relative areas of equilateral triangles whose vertices are on a triangular lattice. - Anton Sherwood (bronto(AT)pobox.com), Apr 05 2001
2 appended to a(n) (for positive n) corresponds to capsomere count in viral architectural structures (cf. A071336). - Lekraj Beedassy, Apr 14 2006
The triangle in A132111 gives the enumeration: n^2 + k*n + k^2, 0 <= k <= n.
The number of coat proteins at each corner of a triangular face of a virus shell. - Parthasarathy Nambi, Sep 04 2007
Numbers of the form (x^2 + y^2 + (x + y)^2)/2. If we let z = - x - y, then all the solutions to x^2 + y^2 + z^2 = k with x + y + z = 0 are k = 2a(n) for any n. - Jon Perry, Dec 16 2012
Sequence of divisors of the hexagonal lattice, except zero (where it is said that an integer n divides a lattice if there exists a sublattice of index n; example: 3 divides the hexagonal lattice). - Jean-Christophe Hervé, May 01 2013
Numbers of the form - (x*y + y*z + x*z) with x + y + z = 0. Numbers of the form x^2 + y^2 + z^2 - (x*y + y*z + x*z) = (x - y)*(x - z) + (y - x) * (y - z) + (z - x) * (z - y). - Michael Somos, Jun 26 2013
Equivalently, the existence spectrum of affine Mendelsohn triple systems, cf. A248107. - David Stanovsky, Nov 25 2014
Lame's solutions to the Helmholtz equation with Dirichlet boundary conditions on the unit-edged equilateral triangle have eigenvalues of the form: (x^2+x*y+y^2)*(4*Pi/3)^2. The actual set, starting at 1 and counting degeneracies, is given by A060428, e.g., the first degeneracy is 49 where (x,y)=(0,7) and (3,5). - Robert Stephen Jones, Oct 01 2015
Curvatures of spheres in one bowl of integers, the Loeschian spheres. Mod 12, numbers equal to 0, 1, 3, 4, 7, 9. - Ed Pegg Jr, Jan 10 2017
Named after the German economist August Lösch (1906-1945). - Amiram Eldar, Jun 10 2021
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REFERENCES
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J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 111.
Ivars Peterson, The Jungles of Randomness: A Mathematical Safari, John Wiley and Sons, (1998) pp. 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Jacob Rus, Flowsnake Earth, Bridges 2017 Conference Proceedings, pp. 237-244.
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FORMULA
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Either n=0 or else in the prime factorization of n all primes of the form 3a+2 must occur to even powers only (there is no restriction of primes congruent to 0 or 1 mod 3).
If n is in the sequence, then n^k is in the sequence (but the converse is not true). n is in the sequence iff n^(2k+1) is in the sequence. - Ray Chandler, Feb 03 2009
The sequence is multiplicative in the sense that if m and n are in the sequence, so is m*n. - Jon Perry, Dec 18 2012
Comments from Richard C. Schroeppel, Jul 20 2016: (Start)
The set is also closed under restricted division: If M and N are members, and M divides N, then N/M is a member.
If N == 2 (mod 3), N is not in the sequence.
The density of members (relative to the integers>0) gradually falls to 0. The density goes as O(1/sqrt(log N)). This implies that, if N is a member, the average expected number of representations of N is O(sqrt(log N)).
Representations usually come in sets of 6: (K,L), (K+L,-K), (K+L,-L) and their negatives. (End)
Since Q(zeta), where zeta is a primitive 3rd root of unity has class number 1, the situation as to whether an integer is of the form x^2 + xy + y^2 is similar to the situation with x^2 + y^2: n is of the that form if and only if every prime p dividing n which is = 5 mod 6 divides it to an even power. The density of 1/sqrt(x) that Rich mentioned is an old result due to Landau. - Victor S. Miller, Jul 20 2016
In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2+xy+y^2=n is 6*Product_{p_i in S_1} (e_i + 1) if all e_j are even and 0 otherwise.
For all Löschian numbers there are nonnegative X,Y such that X^2+XY+Y^2=n. For x,y such that x^2+xy+y^2=n take X=min(|x|,|y|), Y=|x+y| if xy<0 and X=|x|, Y=|y| otherwise. (End)
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MAPLE
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readlib(ifactors): for n from 2 to 200 do m := ifactors(n)[2]: flag := 1: for i from 1 to nops(m) do if m[i, 1] mod 3 = 2 and m[i, 2] mod 2 = 1 then flag := 0; break fi: od: if flag=1 then printf(`%d, `, n) fi: od: # James A. Sellers, Dec 07 2000
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MATHEMATICA
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ok[n_] := Resolve[Exists[{x, y}, Reduce[n == x^2 + x*y + y^2, {x, y}, Integers]]]; Select[Range[0, 192], ok] (* Jean-François Alcover, Apr 18 2011 *)
nn = 14; Select[Union[Flatten[Table[x^2 + x*y + y^2, {x, 0, nn}, {y, 0, x}]]], # <= nn^2 &] (* T. D. Noe, Apr 18 2011 *)
QP = QPochhammer; s = QP[q]^3 / QP[q^3]/3 + O[q]^200; Position[ CoefficientList[s, q], n_ /; n != 0] - 1 // Flatten (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
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PROG
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(Haskell)
import Data.Set (singleton, union, fromList, deleteFindMin)
a003136 n = a003136_list !! (n-1)
a003136_list = f 0 $ singleton 0 where
f x s | m < x ^ 2 = m : f x s'
| otherwise = m : f x'
(union s' $ fromList $ map (\y -> x'^2+(x'+y)*y) [0..x'])
where x' = x + 1
(m, s') = deleteFindMin s
(PARI) isA003136(n)=local(fac, flag); if(n==0, 1, fac=factor(n); flag=1; for(i=1, matsize(fac)[1], if(Mod(fac[i, 1], 3)==2 && Mod(fac[i, 2], 2)==1, flag=0)); flag)
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) \\ Ralf Stephan, Oct 18 2013
(PARI) x='x+O('x^200); p=eta(x)^3/eta(x^3); for(n=0, 199, if(polcoeff(p, n) != 0, print1(n, ", "))) \\ Altug Alkan, Nov 08 2015
(PARI) list(lim)=my(v=List(), y, t); for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y++)); Set(v) \\ Charles R Greathouse IV, Feb 05 2017
(PARI) is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3) \\ Hugo Pfoertner, Aug 04 2023
(Julia)
function isA003136(n)
n % 3 == 2 && return false
n in [0, 1, 3] && return true
M = Int(round(2*sqrt(n/3)))
for y in 0:M, x in 0:y
n == x^2 + y^2 + x*y && return true
end
return false
end
A003136list(upto) = [n for n in 0:upto if isA003136(n)]
(Python)
from itertools import count, islice
from sympy import factorint
def A003136_gen(): return (n for n in count(0) if all(e % 2 == 0 for p, e in factorint(n).items() if p % 3 == 2))
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CROSSREFS
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See A092572 for numbers of form x^2 + 3 y^2 with positive x,y.
See A088534 for the number of representations.
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KEYWORD
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core,easy,nonn,nice
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AUTHOR
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STATUS
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approved
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A007645
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Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).
(Formerly M2637)
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+10
87
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3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613
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OFFSET
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1,1
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COMMENTS
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Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021
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REFERENCES
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D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.
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LINKS
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FORMULA
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p == 0 or 1 (mod 3).
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MAPLE
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select(isprime, [3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014
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MATHEMATICA
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Join[{3}, Select[Prime[Range[150]], Mod[#, 3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)
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PROG
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(Haskell)
a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
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CROSSREFS
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Apart from initial term, same as A045331.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A034020
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Not of the form x^2 + x*y + y^2.
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+10
11
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2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, 22, 23, 24, 26, 29, 30, 32, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 72, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105
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OFFSET
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1,1
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COMMENTS
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Appears to be the sequence of nonsquare n such that sigma(n)==0 (mod 3). - Benoit Cloitre, Sep 17 2002
Numbers n such that n-th coefficient of eta(x)^3/eta(x^3) is zero where eta(x) coefficients are given by A010815. - Benoit Cloitre, Oct 06 2005
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LINKS
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FORMULA
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MATHEMATICA
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Select[Range@ 105, ! Resolve[Exists[{x, y}, Reduce[# == x^2 + x y + y^2, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 06 2016 *)
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PROG
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(PARI) default(seriesprecision, 105); for(n=1, 105, if (polcoeff(eta(x)^3/eta(x^3)+O(x^(n+1)), n) == 0, print1(n, ", "))) \\ Benoit Cloitre, Oct 06 2005
(PARI) x='x+O('x^100); p=eta(x)^3/eta(x^3); for(n=1, 99, if(polcoeff(p, n)==0, print1(n, ", "))); \\ Altug Alkan, Nov 08 2015
(PARI) list(lim)=my(v=List(), y, t); lim\=1; for(x=0, sqrtint(lim\3), my(y=x, t); while((t=x^2+x*y+y^2)<=lim, listput(v, t); y++)); v=Set(v); setminus([2..lim], v) \\ Charles R Greathouse IV, Jul 05 2017
(Haskell)
a034020 n = a034020_list !! (n-1)
a034020_list = f [0..] a003136_list where
f (x:xs) ys'@(y:ys) | x < y = x : f xs ys'
| otherwise = f xs ys
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CROSSREFS
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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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A118886
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Numbers expressible as x^2 + x*y + y^2, 0 <= x <= y, in 2 or more ways.
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+10
11
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49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 651, 676, 679, 703, 721, 741, 763, 777, 784, 793, 817, 819, 868, 871, 889, 903, 931, 949, 961, 973, 988, 1027, 1029, 1036, 1057, 1083, 1092, 1099
(list;
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OFFSET
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1,1
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COMMENTS
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Squares of distances between two points in the triangular lattice in two or more nontrivially different ways.
Numbers whose prime factorization contains at least two (not necessarily distinct) primes congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. Products of two elements of A024606.
If k is in the sequence then so is k * m^2 for m > 0. - David A. Corneth, Jun 21 2018
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2;
a(45) = 931 = 1^2+1*30+30^2 = 9^2+9*25+25^2 = 14^2+14*21+21^2;
a(97) = 1729 = 3^2+3*40+40^2 = 8^2+8*37+37^2 = 15^2+15*32+32^2 = 23^2+23*25+25^2. - Reinhard Zumkeller, Oct 30 2011
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MATHEMATICA
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amax = 2000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
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PROG
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(Haskell)
a118886 n = a118886_list !! (n-1)
a118886_list = filter ((> 1) . a088534) a003136_list
(PARI) is(n)=#bnfisintnorm(bnfinit(z^2+z+1), n) > 2;
select(is, vector(1500, j, j)) \\ Joerg Arndt, Jan 11 2015
(Julia)
function isA118886(n)
n < 49 && return false
n % 3 == 2 && return false
M = Int(round(2*sqrt(n/3)))
count = 0
for y in 0:M, x in 0:y
n == x^2 + y^2 + x*y && (count += 1)
count == 2 && break
end
return count >= 2
end
A118886list(upto) = [n for n in 0:upto if isA118886(n)]
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CROSSREFS
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Subsequence of Loeschian numbers A003136.
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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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A198772
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Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.
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+10
9
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0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 139, 144, 148, 151, 156, 157, 163, 171, 172, 175, 181, 183, 189, 192, 193
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OFFSET
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1,3
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LINKS
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FORMULA
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EXAMPLE
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a(20) = 48 = 4^2 + 4*4 + 4^2, A088534(48) = 1;
a(21) = 52 = 2^2 + 2*6 + 6^2, A088534(52) = 1.
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MATHEMATICA
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amax = 200; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
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PROG
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(Haskell)
a198772 n = a198772_list !! (n-1)
a198772_list = filter ((== 1) . a088534) a003136_list
(Julia)
function isA198772(n)
M = Int(round(2*sqrt(n/3)))
count = 0
for y in 0:M, x in 0:y
n == x^2 + y^2 + x*y && (count += 1)
count == 2 && break
end
return count == 1
end
A198772list(upto) = [n for n in 0:upto if isA198772(n)]
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CROSSREFS
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Subsequence of Loeschian numbers A003136.
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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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A024606
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Numbers of form x^2 + xy + y^2 with distinct x and y > 0.
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+10
8
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7, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 63, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247
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OFFSET
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1,1
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COMMENTS
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Alternatively, numbers expressible in more than one way as i^2 - ij + j^2, where 1 <= i <= j or 1 <= i < j. The following argument shows that the conditions i <= j or i < j are here equivalent. Note first that i^2 - ij + j^2 = (j - i)^2 - (j - i)*j + j^2, so the only non-duplicated values i^2 - ij + j^2 with 1 <= i < j are when j = 2i, whence i^2 - ij + j^2 = 3i^2. On the other hand, the values with i = j are j^2. There are no integer solutions to 3i^2 = j^2 with i >= 1. - Franklin T. Adams-Watters, May 03 2006
Numbers whose prime factorization contains at least one prime congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. - Franklin T. Adams-Watters, May 03 2006
This is a subsequence of Loeschian numbers A003136, closed under multiplication. Its primitive elements are those with exactly one prime factor of form 6k + 1 with multiplicity one (A232436). - Jean-Christophe Hervé, Nov 22 2013
a(1)*a(2)*a(3) = 1729, the Hardy-Ramanujan taxicab number. 1729 is then in the sequence, by the argument of the preceding comment. - Jean-Christophe Hervé, Nov 24 2013
1729 is also the least term that can be written in 4 distinct ways in the given form. Sequence A024614 does not include the restriction x != y, it is the disjoint union of this sequence and A033428 (i.e., 3*x^2) (without 0). - M. F. Hasler, Mar 05 2018
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 7 = 1^2 + 2 + 2^2.
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MATHEMATICA
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Take[Union[Flatten[Table[x^2 + x*y + y^2, {x, 15}, {y, x - 1}]]], 60] (* Robert G. Wilson v, Nov 24 2013 *)
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PROG
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(PARI) for(k=1, 247, my(a088534=sum(x=0, sqrt(k\3), sum(y=max(x, sqrtint(k-x^2)\2), sqrtint(k-2*x^2), x^2+x*y+y^2==k)), a004016d6=sumdiv(k, d, (d%3==1)-(d%3==2))); if(a088534!=a004016d6, print1(k, ", "))) \\ Hugo Pfoertner, Sep 22 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A198773
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Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.
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+10
8
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49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 651, 676, 679, 703, 721, 741, 763, 777, 784, 793, 817, 819, 868, 871, 889, 903, 949, 961, 973, 988, 1027, 1029, 1036, 1057, 1083, 1092, 1099, 1141, 1147
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graph;
refs;
listen;
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text;
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(10) = 273 = 1^2 + 1*16 + 16^2 = 8^2 + 8*11 + 11^2, A088534(273) = 2;
a(11) = 301 = 4^2 + 4*15 + 15^2 = 9^2 + 9*11 + 11^2, A088534(301) = 2.
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MATHEMATICA
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amax = 2000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
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PROG
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(Haskell)
a198773 n = a198773_list !! (n-1)
a198773_list = filter ((== 2) . a088534) a003136_list
(PARI) is(n)=my(t=#bnfisintnorm(bnfinit(z^2+z+1), n)); t==3 || t==4;
select(n->is(n), vector(1500, j, j)) \\ Joerg Arndt, Jan 11 2015
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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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A198774
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Numbers having exactly three representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.
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+10
8
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637, 931, 1183, 1519, 1813, 1911, 2107, 2401, 2527, 2548, 2793, 2989, 3211, 3283, 3549, 3577, 3724, 3871, 4557, 4693, 4732, 4753, 5047, 5239, 5341, 5439, 5733, 6076, 6223, 6253, 6321, 6727, 6811, 7203, 7252, 7267, 7399, 7581, 7644, 7693, 7987, 8379, 8428
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listen;
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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a(1) = 637 = 4^2+4*23+23^2 = 7^2+7*21+21^2 = 12^2+12*17+17^2, A088534(637)=3;
a(2) = 931 = 1^2+1*30+30^2 = 9^2+9*25+25^2 = 14^2+14*21+21^2, A088534(273)=3.
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MATHEMATICA
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amax = 10000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
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PROG
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(Haskell)
a198774 n = a198774_list !! (n-1)
a198774_list = filter ((== 3) . a088534) a003136_list
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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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