OFFSET
1,3
COMMENTS
Arithmetic mean between the number of odd divisors (A001227) and their sum (A000593). This fact was essentially found by the algorithmic search of Jon Maiga's Sequence Machine, and is easily seen to be correct when compared to the PARI-program given by the original author. - Antti Karttunen, Dec 07 2021
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A113415, Sequence Machine.
FORMULA
G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).
a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - Wesley Ivan Hurt, Nov 25 2021 [From PARI prog]
From Antti Karttunen, Dec 07 2021: (Start)
All these formulas, except the last, were found by the Sequence Machine in some form or another:
(End)
MATHEMATICA
Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* Michael De Vlieger, Dec 08 2021 *)
PROG
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 29 2005
STATUS
approved