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A284345 Number of partitions of n into squares dividing n. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

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.

EXAMPLE

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].

MAPLE

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

MATHEMATICA

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}]]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 25 2017

STATUS

approved

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Last modified April 17 21:53 EDT 2021. Contains 343071 sequences. (Running on oeis4.)