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A284345
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Number of partitions of n into squares dividing n.
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8
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1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 6, 1, 3, 1, 6, 1, 1, 1, 7, 2, 1, 4, 8, 1, 1, 1, 15, 1, 1, 1, 27, 1, 1, 1, 11, 1, 1, 1, 12, 6, 1, 1, 28, 2, 3, 1, 14, 1, 7, 1, 15, 1, 1, 1, 16, 1, 1, 8, 46, 1, 1, 1, 18, 1, 1, 1, 114, 1, 1, 4, 20, 1, 1, 1, 66, 11, 1, 1, 22, 1, 1, 1, 23, 1, 11, 1, 24, 1, 1, 1, 91, 1, 3, 12, 67
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OFFSET
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0,5
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences
Index entries for sequences related to sums of squares
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FORMULA
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a(n) = [x^n] Product_{d^2|n} 1/(1 - x^(d^2)).
a(n) = 1 if n is a squarefree.
a(n) = 2 if n is a square of prime.
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EXAMPLE
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a(8) = 3 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are squares {1, 4} therefore we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].
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MAPLE
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with(numtheory):
a:= proc(n) option remember; local b, l; l, b:=
sort(select(issqr, [divisors(n)[]])),
proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
end; b(n, nops(l))
end:
seq(a(n), n=0..100); # Alois P. Heinz, Mar 30 2017
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MATHEMATICA
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Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[Mod[DivisorSigma[0, d[[k]]], 2] == 1] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 100}]]
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CROSSREFS
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Cf. A000290, A001156, A018818, A046951, A066882, A161148, A225244, A284289.
Sequence in context: A222580 A316978 A331023 * A337619 A183214 A101221
Adjacent sequences: A284342 A284343 A284344 * A284346 A284347 A284348
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Mar 25 2017
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STATUS
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approved
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