A284233 Sum of odd prime power divisors of n (not including 1).

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Author

Ilya Gutkovskiy, Mar 23 2017

Keywords

• nonn
• easy

Examples

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.
		

Links

• Eric Weisstein's World of Mathematics, Prime Power.
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

Programs

Mathematica

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

Formula

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).

a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.

a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.

a(2^k*p) = p for p is a prime > 2.

a(2^k) = 0.

Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024