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A284839 Number of compositions (ordered partitions) of n into prime power divisors of n (including 1). 1
1, 1, 2, 2, 6, 2, 24, 2, 56, 20, 128, 2, 1490, 2, 741, 449, 5272, 2, 36901, 2, 81841, 3320, 29966, 2, 4135004, 572, 200389, 26426, 5452795, 2, 110187694, 2, 47350056, 226019, 9262156, 51885, 10783889706, 2, 63346597, 2044894, 14064551462, 2, 109570982403, 2, 35537376325, 470326038, 2972038874, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Prime Power

Index entries for sequences related to compositions

FORMULA

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

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

EXAMPLE

a(4) = 6 because 4 has 3 divisors {1, 2, 4} and all are prime powers therefore we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

MAPLE

with(numtheory):

a:= proc(n) local d, b; d, b:= select(x->

      nops(factorset(x))<2, divisors(n)),

      proc(n) option remember; `if`(n=0, 1,

        add(`if`(j>n, 0, b(n-j)), j=d))

      end: b(n)

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Apr 15 2017

MATHEMATICA

Table[d = Divisors[n]; Coefficient[Series[1/(1 - x - Sum[Boole[PrimePowerQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 47}]

CROSSREFS

Cf. A000961, A066882, A100346, A284465.

Sequence in context: A324158 A196441 A130674 * A286376 A100346 A306387

Adjacent sequences:  A284836 A284837 A284838 * A284840 A284841 A284842

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 03 2017

STATUS

approved

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Last modified April 6 15:23 EDT 2020. Contains 333276 sequences. (Running on oeis4.)