|
%I #28 Dec 09 2021 01:24:29
%S 1,1,3,1,4,3,5,1,8,4,7,3,8,5,14,1,10,8,11,4,18,7,13,3,17,8,22,5,16,14,
%T 17,1,26,10,26,8,20,11,30,4,22,18,23,7,42,13,25,3,30,17,38,8,28,22,38,
%U 5,42,16,31,14,32,17,55,1,44,26,35,10,50,26,37,8,38,20,65,11,50,30,41
%N Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.
%C 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
%H Antti Karttunen, <a href="/A113415/b113415.txt">Table of n, a(n) for n = 1..20000</a>
%H Jon Maiga, <a href="http://sequencedb.net/s/A113415">Computer-generated formulas for A113415</a>, Sequence Machine.
%F G.f.: Sum_{k>0} x^k/(1-x^(2k))^2 = Sum_{k>0} k x^(2k-1)/(1-x^(2k-1)).
%F a(n) = (1/2) * Sum_{d|n} (d+1)*(d mod 2). - _Wesley Ivan Hurt_, Nov 25 2021 [From PARI prog]
%F From _Antti Karttunen_, Dec 07 2021: (Start)
%F All these formulas, except the last, were found by the Sequence Machine in some form or another:
%F a(n) = (1/2) * (A000593(n)+A001227(n)).
%F a(n) = A069734(A000265(n)). [See either Rutherford's or Luschny's formula in A069734]
%F a(n) = A349371(n) / A001511(n).
%F a(n) = A349371(A000265(n)) = A336840(A064989(n)).
%F a(n) = a(2*n) = a(A000265(n)) = A349916(4*n).
%F (End)
%t Array[DivisorSum[#, If[OddQ[#], (# + 1)/2, 0] &] &, 79] (* _Michael De Vlieger_, Dec 08 2021 *)
%o (PARI) a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)))
%Y Cf. A000265, A000593, A001227, A001511, A003602, A048673, A064989, A069734, A336840, A349371, A349915 (Dirichlet inverse).
%Y Quadrisection of A349916.
%K nonn
%O 1,3
%A _Michael Somos_, Oct 29 2005
|
