%I #12 Jul 25 2024 03:12:47

%S 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,

%T 29,8,31,0,14,17,12,12,37,19,16,5,41,10,43,11,17,23,47,3,56,30,20,13,

%U 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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.

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

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

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

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

%F a(2^k) = 0.

%F 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

%e 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.

%t 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]]

%t Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]

%t 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 *)

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

%K nonn,easy

%O 1,3

%A _Ilya Gutkovskiy_, Mar 23 2017