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A087153
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Number of partitions of n into nonsquares.
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27
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1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 9, 13, 15, 20, 24, 30, 37, 47, 55, 71, 83, 103, 123, 151, 178, 218, 257, 310, 366, 440, 515, 617, 722, 857, 1003, 1184, 1380, 1625, 1889, 2214, 2570, 3000, 3472, 4042, 4669, 5414, 6244, 7221, 8303, 9583, 10998, 12655, 14502
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OFFSET
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0,6
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COMMENTS
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Also, number of partitions of n where there are fewer than k parts equal to k for all k. - Jon Perry and Vladeta Jovovic, Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+2.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The Heinz numbers of the integer partitions described in Perry and Jovovic's comment are given by A325128, while the Heinz numbers of the integer partitions described in the name are given by A325129. In the former case, the first 10 terms count the following integer partitions:
() (2) (3) (4) (5) (6) (7) (8) (9)
(32) (33) (43) (44) (54)
(42) (52) (53) (63)
(62) (72)
(332) (432)
while in the latter case they count the following:
() (2) (3) (22) (5) (6) (7) (8) (63)
(32) (33) (52) (53) (72)
(222) (322) (62) (333)
(332) (522)
(2222) (3222)
(End)
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REFERENCES
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G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. See page 48.
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LINKS
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FORMULA
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G.f.: Product_{i>=1} (Sum_{j=0..i-1} x^(i*j)). - Jon Perry, Jul 26 2004
a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) * sqrt(Pi) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Dec 30 2016
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EXAMPLE
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n=7: 2+5 = 2+2+3 = 7: a(7)=3;
n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.
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MAPLE
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g:=product((1-x^(i^2))/(1-x^i), i=1..70):gser:=series(g, x=0, 60):seq(coeff(gser, x^n), n=1..53); # Emeric Deutsch, Feb 09 2006
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MATHEMATICA
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nn=54; CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, nn}], {x, 0, nn}], x] (* Robert G. Wilson v, Aug 05 2004 *)
nmax = 100; CoefficientList[Series[Product[(1 - x^(k^2))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 29 2016 *)
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PROG
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(Haskell)
a087153 = p a000037_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(PARI) first(n)=my(x='x+O('x^(n+1))); Vec(prod(m=1, sqrtint(n), (1-x^m^2)/(1-x^m))*prod(m=sqrtint(n)+1, n, 1/(1-x^m))) \\ Charles R Greathouse IV, Aug 28 2016
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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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