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URL: https://oeis.org/A284839

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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
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 03 2017
STATUS
approved