OFFSET
1,9
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Odd Divisor Function
FORMULA
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)).
a(n) = Sum_{d|n, d odd nonprime} d.
a(A093641(n)) = 1.
EXAMPLE
a(9) = 10 because 9 has 3 divisors {1, 3, 9} among which 2 are odd nonprime {1, 9} therefore 1 + 9 = 10.
MAPLE
with(numtheory):
a:= n-> add(`if`(d::even or d::prime, 0, d), d=divisors(n)):
seq(a(n), n=1..100); # Alois P. Heinz, Jan 18 2017
MATHEMATICA
Table[DivisorSum[n, #1 &, Mod[#1, 2] == 1 && ! PrimeQ[#1] &], {n, 97}]
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]]
PROG
(PARI) a(n) = sumdiv(n, d, !isprime(d)*(d%2)*d); \\ Michel Marcus, Sep 18 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 17 2017
STATUS
approved