Search: a011757 -id:a011757
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5, 16, 30, 44, 54, 76, 84, 108, 122, 120, 166, 182, 184, 234, 192, 260, 264, 294, 304, 342, 378, 342, 408, 426, 414, 468, 488, 474, 516, 576, 588, 576, 604, 590, 696, 694, 728, 694, 756, 828, 774, 776, 870, 862, 852, 1010, 922, 998, 916, 1020, 1032, 1110
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OFFSET
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1,1
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COMMENTS
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Does a(n) = k^2 have infinitely many solutions? E.g., 16 = 4^2; 576 = 24^2; ...
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LINKS
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MATHEMATICA
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PROG
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(PARI) j=[]; for(n=1, 150, j=concat(j, prime((n+1)^2)-prime(n^2))); j
(PARI) { default(primelimit, 2*10^7); for (n=1, 1000, write("b063076.txt", n, " ", prime((n+1)^2) - prime(n^2)) ) } \\ Harry J. Smith, Aug 17 2009
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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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A001156
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Number of partitions of n into squares.
(Formerly M0221 N0079)
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+10
110
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1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, 14, 14, 16, 19, 20, 21, 23, 26, 27, 28, 31, 34, 37, 38, 43, 46, 49, 50, 55, 60, 63, 66, 71, 78, 81, 84, 90, 98, 104, 107, 116, 124, 132, 135, 144, 154, 163, 169, 178, 192, 201, 209, 220, 235, 247, 256
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OFFSET
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0,5
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COMMENTS
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Number of partitions of n such that number of parts equal to k is multiple of k for all k. - Vladeta Jovovic, Aug 01 2004
Of course p_{4*square}(n)>0. In fact p_{4*square}(32n+28)=3 times p_{4*square}(8n+7) and p_{4*square}(72n+69) is even. These seem to be the only arithmetic properties the function p_{4*square(n)} possesses. Similar results hold for partitions into positive squares, distinct squares and distinct positive squares. - Michael David Hirschhorn, May 05 2005
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REFERENCES
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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).
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LINKS
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J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Partition
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FORMULA
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G.f.: Product_{m>=1} 1/(1-x^(m^2)).
G.f.: Sum_{n>=0} x^(n^2) / Product_{k=1..n} (1 - x^(k^2)). - Paul D. Hanna, Mar 09 2012
a(n) = f(n,1,3) with f(x,y,z) = if x<y then 0^x else f(x-y,y,z)+f(x,y+z,z+2). - Reinhard Zumkeller, Nov 08 2009
Conjecture (Jan Bohman, Carl-Erik Fröberg, Hans Riesel, 1979): a(n) ~ c * n^(-alfa) * exp(beta*n^(1/3)), where c = 1/18.79656, beta = 3.30716, alfa = 1.16022. - Vaclav Kotesovec, Aug 19 2015
Correct values of these constants are:
1/c = sqrt(3) * (4*Pi)^(7/6) / Zeta(3/2)^(2/3) = 17.49638865935104978665...
alfa = 7/6 = 1.16666666666666666...
beta = 3/2 * (Pi/2)^(1/3) * Zeta(3/2)^(2/3) = 3.307411783596651987...
a(n) ~ 3^(-1/2) * (4*Pi*n)^(-7/6) * Zeta(3/2)^(2/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)). [Hardy & Ramanujan, 1917]
(End)
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EXAMPLE
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p_{4*square}(23)=1 because 23 = 3^2 + 3^2 + 2^2 + 1^2 and there is no other partition of 23 into squares.
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 +...
A(x) = 1/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)*(1-x^25)*...)
A(x) = 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^9/((1-x)*(1-x^4)*(1-x^9)) + x^16/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)) + ...
The a(14) = 6 integer partitions into squares are:
(941)
(911111)
(44411)
(44111111)
(41111111111)
(11111111111111)
while the a(14) = 6 integer partitions in which the multiplicity of k is a multiple of k for all k are:
(333221)
(33311111)
(22222211)
(2222111111)
(221111111111)
(11111111111111)
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
end:
a:= n-> b(n, isqrt(n)):
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MATHEMATICA
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CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
Join[{1}, Table[Length@PowersRepresentations[n, n, 2], {n, 68}]] (* Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt@ n^2]; Array[f, 67] (* Robert G. Wilson v, Apr 14 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n, Sqrt[n]//Floor]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)
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PROG
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(Haskell)
a001156 = p (tail a000290_list) where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(PARI) {a(n)=polcoeff(1/prod(k=1, sqrtint(n+1), 1-x^(k^2)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 09 2012
(PARI) {a(n)=polcoeff(1+sum(m=1, sqrtint(n+1), x^(m^2)/prod(k=1, m, 1-x^(k^2)+x*O(x^n))), n)} \\ Paul D. Hanna, Mar 09 2012
(Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^(k^2)): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 11 2018
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CROSSREFS
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Cf. A001462, A003114, A006141, A011757, A039900, A047993, A052335, A062457, A064174, A078135, A109298, A117144.
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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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More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006
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STATUS
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approved
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A055875
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a(0)=1, a(n) = prime(n^3).
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+10
20
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1, 2, 19, 103, 311, 691, 1321, 2309, 3671, 5519, 7919, 10957, 14753, 19403, 24809, 31319, 38873, 47657, 57559, 69031, 81799, 96137, 112291, 130073, 149717, 171529, 195043, 220861, 248851, 279431, 312583, 347707, 386093, 427169, 470933, 517553
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OFFSET
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0,2
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COMMENTS
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A sequence of increments for Shell sort that produces good results. A bit better than Sedgewick's A036562 and A003462.
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LINKS
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FORMULA
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MATHEMATICA
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PROG
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(PARI) first(n) = { my(res = vector(n), t = 0); forprime(p = 2, oo, t++; if(ispower(t, 3, &i), print1([i, p]", "); res[i] = p; if(i >= n, return(concat(1, res))))) } \\ David A. Corneth, Apr 13 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Steven Pigeon (pigeon(AT)iro.umontreal.ca), Jul 14 2000
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EXTENSIONS
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STATUS
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approved
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A324588
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Heinz numbers of integer partitions of n into perfect squares (A001156).
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+10
11
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1, 2, 4, 7, 8, 14, 16, 23, 28, 32, 46, 49, 53, 56, 64, 92, 97, 98, 106, 112, 128, 151, 161, 184, 194, 196, 212, 224, 227, 256, 302, 311, 322, 343, 368, 371, 388, 392, 419, 424, 448, 454, 512, 529, 541, 604, 622, 644, 661, 679, 686, 736, 742, 776, 784, 827, 838
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OFFSET
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1,2
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A011757.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
7: {4}
8: {1,1,1}
14: {1,4}
16: {1,1,1,1}
23: {9}
28: {1,1,4}
32: {1,1,1,1,1}
46: {1,9}
49: {4,4}
53: {16}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
92: {1,1,9}
97: {25}
98: {1,4,4}
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MATHEMATICA
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Select[Range[100], And@@Cases[FactorInteger[#], {p_, _}:>IntegerQ[Sqrt[PrimePi[p]]]]&]
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CROSSREFS
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Cf. A001156, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A118914, A276078.
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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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A123914
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a(n) = prime(n)^2 - prime(n^2). Commutator of (primes, squares) at n.
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+10
9
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2, 2, 2, -4, 24, 18, 62, 50, 110, 300, 300, 542, 672, 656, 782, 1190, 1602, 1578, 2052, 2300, 2246, 2780, 3086, 3710, 4772, 5150, 5090, 5442, 5400, 5772, 8556, 9000, 10032, 9980, 12270, 12174, 13328, 14520, 15146, 16430, 17714, 17660, 20604, 20502, 21200
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OFFSET
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1,1
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COMMENTS
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a(4) = -4 is the only negative value. All values are even. Asymptotically a(n) ~ (n log n)^2 - (n^2) log (n^2) = (n^2)*(log n)^2 - 2*(n^2)*(log n) = (n^2)*((log n)^2 - 2*log n) = O((n^2)*(log n)^2) which is the same as the asymptotic of commutator [primes, triangular numbers] at n, or, for that matter, commutator [primes, k-tonal numbers] at n for any k > 2.
Proof that a(n) > 0 for n <> 4: It is known that pi(k^2) >= pi(k)^2 for k <> 7 (see A291440). Take k = prime(n) to get pi(prime(n)^2) >= pi(prime(n))^2 = n^2 for prime(n) <> 7 = prime(4). Thus for n <> 4 there are at least n^2 primes <= prime(n)^2, so prime(n^2) <= prime(n)^2, implying a(n) >= 0. But a prime cannot equal a square, so a(n) > 0 for n <> 4. - Jonathan Sondow, Nov 04 2017
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REFERENCES
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LINKS
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FORMULA
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a(n) = square(prime(n)) - prime(square(n)).
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EXAMPLE
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a(1) = prime(1)^2 - prime(1^2) = prime(1)^2 - prime(1^2) = 4 - 2 = 2.
a(2) = prime(2)^2 - prime(2^2) = prime(2)^2 - prime(2^2) = 9 - 7 = 2.
a(3) = prime(3)^2 - prime(3^2) = prime(3)^2 - prime(3^2) = 25 - 23 = 2.
a(4) = prime(4)^2 - prime(4^2) = prime(4)^2 - prime(4^2) = 49 - 53 = -4.
a(5) = prime(5)^2 - prime(5^2) = prime(5)^2 - prime(5^2) = 121 - 97 = 24.
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MATHEMATICA
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Table[(Prime[n])^2 - Prime[n^2], {n, 1, 300}] (* G. C. Greubel, Sep 15 2015 *)
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PROG
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(Magma) [NthPrime(n)^2 - NthPrime(n^2): n in [1..60]]; // Vincenzo Librandi, Sep 16 2015
(PARI) vector(100, n, prime(n)^2 - prime(n^2)) \\ Altug Alkan, Oct 05 2015
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A324587
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Heinz numbers of integer partitions of n into distinct perfect squares (A033461).
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+10
8
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1, 2, 7, 14, 23, 46, 53, 97, 106, 151, 161, 194, 227, 302, 311, 322, 371, 419, 454, 541, 622, 661, 679, 742, 827, 838, 1009, 1057, 1082, 1193, 1219, 1322, 1358, 1427, 1589, 1619, 1654, 1879, 2018, 2114, 2143, 2177, 2231, 2386, 2437, 2438, 2741, 2854, 2933
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OFFSET
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1,2
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A011757.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
2: {1}
7: {4}
14: {1,4}
23: {9}
46: {1,9}
53: {16}
97: {25}
106: {1,16}
151: {36}
161: {4,9}
194: {1,25}
227: {49}
302: {1,36}
311: {64}
322: {1,4,9}
371: {4,16}
419: {81}
454: {1,49}
541: {100}
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MATHEMATICA
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Select[Range[1000], And@@Cases[FactorInteger[#], {p_, k_}:>k==1&&IntegerQ[Sqrt[PrimePi[p]]]]&]
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CROSSREFS
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Cf. A001156, A005117, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A276078.
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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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A351982
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Number of integer partitions of n into prime parts with prime multiplicities.
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+10
8
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1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 1, 3, 3, 3, 0, 1, 4, 5, 5, 3, 3, 5, 8, 5, 5, 6, 8, 8, 11, 7, 8, 10, 17, 14, 14, 12, 17, 17, 21, 18, 23, 20, 28, 27, 31, 27, 36, 32, 35, 37, 46, 41, 52, 45, 60, 58, 63, 59, 78, 71, 76, 81, 87, 80, 103, 107, 113, 114, 127
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OFFSET
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0,7
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LINKS
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EXAMPLE
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The partitions for n = 4, 6, 10, 19, 20, 25:
(22) (33) (55) (55333) (7733) (55555)
(222) (3322) (55522) (77222) (77722)
(22222) (3333322) (553322) (5533333)
(33322222) (5522222) (5553322)
(332222222) (55333222)
(55522222)
(333333322)
(3333322222)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]], {n, 0, 30}]
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CROSSREFS
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The version for just prime multiplicities is A055923, ranked by A056166.
These partitions are ranked by A346068.
A001221 counts constant partitions of prime length, ranked by A053810.
A038499 counts partitions of prime length.
A101436 counts parts of prime signature that are themselves prime.
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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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A109770
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Prime(1^2) + prime(2^2) + ... + prime(n^2).
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+10
5
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2, 9, 32, 85, 182, 333, 560, 871, 1290, 1831, 2492, 3319, 4328, 5521, 6948, 8567, 10446, 12589, 15026, 17767, 20850, 24311, 28114, 32325, 36962, 42013, 47532, 53539, 60020, 67017, 74590, 82751, 91488, 100829, 110760, 121387, 132708, 144757
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 2 = prime(1^2).
a(2) = 9 = 2 + 7 = prime(1^2) + prime(2^2).
a(3) = 32 = 2 + 7 + 23 = prime(1^2) + prime(2^2) + prime(3^2).
a(4) = 32 = 2 + 7 + 23 + 53 = prime(1^2) + prime(2^2) + prime(3^2) prime(4^2).
a(x) = 2+7+23+53+97+151+227+311+419+541+661+827+1009+1193+1427+1619+1879+2143+2437+2741+3083+3461+3803+4211+4637+5051+5519+6007+6481+6997+7573+8161+8737+9341+9931+10627+11321+12049+12743+13499+14327 = 185326.
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MATHEMATICA
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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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2, 53, 419, 1619, 4637, 10627, 21391, 38873, 65687, 104729, 159521, 233879, 331943, 459341, 620201, 821641, 1069603, 1370099, 1731659, 2160553, 2667983, 3260137, 3948809, 4742977, 5653807, 6691987, 7867547, 9195889, 10688173, 12358069
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OFFSET
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1,1
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COMMENTS
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Since the prime number theorem is the statement that prime[n] ~ n * log n as n -> infinity [Hardy and Wright, page 10] we have a(n) = prime(n^4) is asymptotically (n^4)*log(n^4) = 4*(n^4)*log(n).
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = prime(1^4) = 2,
a(2) = prime(2^4) = 53,
a(3) = prime(3^4) = 419, etc.
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MATHEMATICA
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PROG
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(Sage) [nth_prime(n^4) for n in (1..30)] # G. C. Greubel, Dec 09 2018
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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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A323526
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One and prime numbers whose prime index is a perfect square.
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+10
4
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1, 2, 7, 23, 53, 97, 151, 227, 311, 419, 541, 661, 827, 1009, 1193, 1427, 1619, 1879, 2143, 2437, 2741, 3083, 3461, 3803, 4211, 4637, 5051, 5519, 6007, 6481, 6997, 7573, 8161, 8737, 9341, 9931, 10627, 11321, 12049, 12743, 13499, 14327, 15101, 15877, 16747, 17609
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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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MATHEMATICA
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Array[Prime[#^2]&, 20]
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PROG
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(PARI) vector(50, n, if (n==1, 1, prime((n-1)^2))) \\ Michel Marcus, Feb 15 2019
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CROSSREFS
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Cf. A000290, A000720, A001248, A011757, A026478, A056239, A323520, A323521, A323525, A323527, A323528.
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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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