A284117 Sum of proper prime power divisors of n.

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

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).

Additive with a(p^e) = (p^(e+1)-1)/(p-1) - p - 1. - Amiram Eldar, Jul 24 2024

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}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

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 A364103

Adjacent sequences: A284114 A284115 A284116 * A284118 A284119 A284120

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Mar 20 2017

Status

approved