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A284117 Sum of proper prime power divisors of n.

%I A284117 #22 Jul 24 2024 21:31:08
%S A284117 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,
%T A284117 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,
%U A284117 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
%N A284117 Sum of proper prime power divisors of n.
%H A284117 Robert Israel, <a href="/A284117/b284117.txt">Table of n, a(n) for n = 1..10000</a>
%H A284117 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A284117 G.f.: Sum_{p prime, k>=2} p^k*x^(p^k)/(1 - x^(p^k)).
%F A284117 a(n) = Sum_{d|n, d = p^k, p prime, k >= 2} d.
%F A284117 a(n) = 0 if n is a squarefree (A005117).
%F A284117 Additive with a(p^e) = (p^(e+1)-1)/(p-1) - p - 1. - _Amiram Eldar_, Jul 24 2024
%e A284117 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.
%p A284117 f:= n -> add(t[1]*(t[1]^t[2]-t[1])/(t[1]-1), t=ifactors(n)[2]):
%p A284117 map(f, [$1..100]); # _Robert Israel_, Mar 31 2017
%t A284117 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]]
%t A284117 Table[Total[Select[Divisors[n], PrimePowerQ[#1] && PrimeOmega[#1] > 1 &]], {n, 100}]
%t A284117 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 *)
%o A284117 (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
%o A284117 (PARI) a(n) = sumdiv(n, d, d*(isprimepower(d) && !isprime(d))); \\ _Michel Marcus_, Apr 01 2017
%Y A284117 Cf. A001414, A008472, A023888, A023889, A046660, A246547.
%K A284117 nonn,easy
%O A284117 1,4
%A A284117 _Ilya Gutkovskiy_, Mar 20 2017