OFFSET
1,4
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Mar 20 2017
STATUS
approved