A284345 Number of partitions of n into squares dividing n.

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

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: A376140 A316978 A331023 * A347463 A373378 A337619

Adjacent sequences: A284342 A284343 A284344 * A284346 A284347 A284348

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 25 2017

Status

approved