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A284289 Number of partitions of n into prime power divisors of n (not including 1). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

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.

EXAMPLE

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].

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 *)

CROSSREFS

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

Sequence in context: A166974 A292504 A281118 * A111588 A070972 A180229

Adjacent sequences:  A284286 A284287 A284288 * A284290 A284291 A284292

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 24 2017

STATUS

approved

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Last modified June 13 20:25 EDT 2021. Contains 345009 sequences. (Running on oeis4.)