cp's OEIS Frontend

A279051 Sum of odd nonprime divisors of n.

%I A279051 #21 Feb 16 2025 08:33:37
%S A279051 1,1,1,1,1,1,1,1,10,1,1,1,1,1,16,1,1,10,1,1,22,1,1,1,26,1,37,1,1,16,1,
%T A279051 1,34,1,36,10,1,1,40,1,1,22,1,1,70,1,1,1,50,26,52,1,1,37,56,1,58,1,1,
%U A279051 16,1,1,94,1,66,34,1,1,70,36,1,10,1,1,116,1,78,40,1,1,118,1,1,22,86,1,88,1,1,70,92,1,94,1,96,1,1
%N A279051 Sum of odd nonprime divisors of n.
%H A279051 G. C. Greubel, <a href="/A279051/b279051.txt">Table of n, a(n) for n = 1..5000</a>
%H A279051 Ilya Gutkovskiy, <a href="/A279051/a279051.pdf">Extended graphical example</a>
%H A279051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>
%H A279051 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A279051 G.f.: A(x) = B(x) - C(x), where B(x) = Sum_{k>=1} k*x^k/(1 + x^k), C(x) = Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).
%F A279051 a(n) = Sum_{d|n, d odd nonprime} d.
%F A279051 a(A093641(n)) = 1.
%e A279051 a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.
%p A279051 with(numtheory):
%p A279051 a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):
%p A279051 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jan 18 2017
%t A279051 Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]
%t A279051 nmax = 97; Rest[CoefficientList[Series[Sum[k x^k/(1 + x^k), {k, 1, nmax}] - Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]
%o A279051 (PARI) a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ _Michel Marcus_, Sep 18 2017
%Y A279051 Cf. A000593, A014076, A023890, A005069, A033272, A093641.
%K A279051 nonn
%O A279051 1,9
%A A279051 _Ilya Gutkovskiy_, Jan 17 2017