A284289 - OEIS

%I #11 Feb 16 2025 08:33:43

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

%H Alois P. Heinz, <a href="/A284289/b284289.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

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

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

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

%e 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 with(numtheory):

%p a:= proc(n) option remember; local b, l; l, b:= sort(

%p       [select(x-> nops(ifactors(x)[2])=1, divisors(n))[]]),

%p       proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,

%p         b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))

%p       end; b(n, nops(l))

%p     end:

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 30 2017

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

%K nonn,changed

%O 0,5

%A _Ilya Gutkovskiy_, Mar 24 2017