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

%I A284289 #11 Feb 16 2025 08:33:43
%S A284289 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,
%T A284289 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,
%U A284289 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
%N A284289 Number of partitions of n into prime power divisors of n (not including 1).
%H A284289 Alois P. Heinz, <a href="/A284289/b284289.txt">Table of n, a(n) for n = 0..10000</a>
%H A284289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%H A284289 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A284289 a(n) = [x^n] Product_{p^k|n, p prime, k >= 1} 1/(1 - x^(p^k)).
%F A284289 a(n) = 1 if n is a prime.
%F A284289 a(n) = 2 if n is a semiprime.
%e A284289 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].
%p A284289 with(numtheory):
%p A284289 a:= proc(n) option remember; local b, l; l, b:= sort(
%p A284289       [select(x-> nops(ifactors(x)[2])=1, divisors(n))[]]),
%p A284289       proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A284289         b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A284289       end; b(n, nops(l))
%p A284289     end:
%p A284289 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 30 2017
%t A284289 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 *)
%t A284289 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 *)
%Y A284289 Cf. A014652, A018818, A023894, A066882, A225244, A211110, A246655.
%K A284289 nonn
%O A284289 0,5
%A A284289 _Ilya Gutkovskiy_, Mar 24 2017