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A284117 Sum of proper prime power divisors of n. 3
0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 13, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 21, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 29 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of divisors

FORMULA

G.f.: Sum_{p prime, k>=2} p^k*x^(p^k)/(1 - x^(p^k)).

a(n) = Sum_{d|n, d = p^k, p prime, k >= 2} d.

a(n) = 0 if n is a squarefree (A005117).

EXAMPLE

a(8) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 2 are proper prime powers {4, 8} therefore 4 + 8 = 12.

MAPLE

f:= n -> add(t[1]*(t[1]^t[2]-t[1])/(t[1]-1), t=ifactors(n)[2]):

map(f, [$1..100]); # Robert Israel, Mar 31 2017

MATHEMATICA

nmax = 100; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

Table[Total[Select[Divisors[n], PrimePowerQ[#1] && PrimeOmega[#1] > 1 &]], {n, 100}]

PROG

(PARI) concat([0, 0, 0], Vec(sum(k=1, 100, (isprimepower(k) && bigomega(k)>1) * k * x^k/(1 - x^k)) + O(x^101))) \\ Indranil Ghosh, Mar 21 2017

(PARI) a(n) = sumdiv(n, d, d*(isprimepower(d) && !isprime(d))); \\ Michel Marcus, Apr 01 2017

CROSSREFS

Cf. A001414, A008472, A023888, A023889, A046660, A246547.

Sequence in context: A242707 A236379 A126849 * A183099 A162296 A169773

Adjacent sequences:  A284114 A284115 A284116 * A284118 A284119 A284120

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 20 2017

STATUS

approved

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Last modified June 5 14:48 EDT 2020. Contains 334841 sequences. (Running on oeis4.)