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

1,3

Links

Table of n, a(n) for n=1..80.

Eric Weisstein's World of Mathematics, Prime Power.

Index entries for sequences related to sums of divisors.

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A284599 A005069 A366840 * A326990 A037284 A225058

Adjacent sequences: A284230 A284231 A284232 * A284234 A284235 A284236

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Mar 23 2017

Status

approved