A279051 - OEIS

%I #20 Sep 18 2017 02:53:47

%S 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 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 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 Sum of odd nonprime divisors of n.

%H G. C. Greubel, <a href="/A279051/b279051.txt">Table of n, a(n) for n = 1..5000</a>

%H Ilya Gutkovskiy, <a href="/A279051/a279051.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>

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

%F 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 a(n) = Sum_{d|n, d odd nonprime} d.

%F a(A093641(n)) = 1.

%e a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.

%p with(numtheory):

%p a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Jan 18 2017

%t Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]

%t 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 (PARI) a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ _Michel Marcus_, Sep 18 2017

%Y Cf. A000593, A014076, A023890, A005069, A033272, A093641.

%K nonn

%O 1,9

%A _Ilya Gutkovskiy_, Jan 17 2017