A284289 Number of partitions of n into prime power divisors of n (not including 1).

1, 0, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 7, 1, 2, 2, 10, 1, 7, 1, 10, 2, 2, 1, 34, 2, 2, 5, 13, 1, 21, 1, 36, 2, 2, 2, 72, 1, 2, 2, 73, 1, 28, 1, 19, 13, 2, 1, 249, 2, 10, 2, 22, 1, 50, 2, 127, 2, 2, 1, 419, 1, 2, 17, 202, 2, 42, 1, 28, 2, 43, 1, 1260, 1, 2, 13, 31, 2, 49, 1, 801, 23, 2, 1, 774, 2, 2, 2, 280, 1, 608

Offset: 0

Author

Ilya Gutkovskiy, Mar 24 2017

Keywords

• nonn

Examples

			a(8) = 4 because 8 has 4 divisors {1, 2, 4, 8} among which 3 are prime powers {2, 4, 8} therefore we have [8], [4, 4], [4, 2, 2] and [2, 2, 2, 2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Prime Power
• Index entries for related partition-counting sequences

Crossrefs

Cf. A014652, A018818, A023894, A066882, A225244, A211110, A246655.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> nops(ifactors(x)[2])=1, 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

Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[PrimePowerQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 0, 90}] (* or *)
a[0]=1; a[1]=0; a[n_] := Length@IntegerPartitions[n, All, Join @@ (#[[1]]^Range[#[[2]]] & /@ FactorInteger[n])]; a /@ Range[0, 90] (* Giovanni Resta, Mar 25 2017 *)

Formula

a(n) = [x^n] Product_{p^k|n, p prime, k >= 1} 1/(1 - x^(p^k)).

a(n) = 1 if n is a prime.

a(n) = 2 if n is a semiprime.