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A098985 Denominators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).

%I A098985 #23 Jun 13 2022 12:36:54
%S A098985 1,1,1,3,1,5,3,7,1,9,5,11,3,13,7,5,1,17,9,19,5,21,11,23,3,25,13,27,7,
%T A098985 29,5,31,1,11,17,35,9,37,19,39,5,41,21,43,11,15,23,47,3,49,25,17,13,
%U A098985 53,27,55,7,57,29,59,5,61,31,63,1,65,11,67,17,23,35,71,9,73,37,75,19,77,39,79
%N A098985 Denominators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).
%C A098985 For n>0, a(n) is the denominator of Sum_{odd d|n} 1/d. See Sumit Kumar Jha link. - _Michel Marcus_, Jul 21 2020
%H A098985 Sumit Kumar Jha, <a href="https://arxiv.org/abs/2007.04243">An identity for the sum of inverses of odd divisors of n in terms of the number of representations of n as a sum of r squares</a>, arXiv:2007.04243 [math.GM], 2020.
%e A098985 2*q-2*q^2+8/3*q^3-2*q^4+12/5*q^5-8/3*q^6+16/7*q^7-2*q^8+26/9*q^9-...
%p A098985 A098985_list := proc(n::integer)
%p A098985     local q,m,nsq ;
%p A098985     nsq := floor(sqrt(n)) ;
%p A098985     add(q^(m^2),m=-nsq-1..nsq+1) ;
%p A098985     log(%) ;
%p A098985     taylor(%,q=0,n+1) ;
%p A098985     [seq( denom(coeftayl(%,q=0,i)) ,i=1..n) ] ;
%p A098985 end proc:
%p A098985 A098985_list(200) ; # _R. J. Mathar_, Jul 16 2020
%p A098985 A336114 := proc(n::integer)
%p A098985     local a ;
%p A098985     for d in numtheory[divisors](n) do
%p A098985         if type(d,'odd') then
%p A098985             a := a+1/d ;
%p A098985         end if;
%p A098985     end do;
%p A098985     denom(a) ;
%p A098985 end proc:
%p A098985 seq(A336114(n),n=1..70) ; # _R. J. Mathar_, Jul 16 2020
%t A098985 Denominator[CoefficientList[Series[Log[Sum[q^m^2, {m, -Infinity, Infinity}]], {q, 0, 79}], q]] (* _L. Edson Jeffery_, Jul 14 2014 *)
%t A098985 a[n_] := Denominator @ DivisorSum[n, 1/# &, OddQ[#] &]; Array[a, 100] (* _Amiram Eldar_, Jul 09 2020 *)
%o A098985 (PARI) lista(nn) = {my(k = sqrtint(nn), s = sum(m=-k-1, k+1, x^(m^2)) + O(x^nn)); apply(x->denominator(x), Vec(log(s)));} \\ _Michel Marcus_, Jul 17 2020
%o A098985 (PARI) a(n) = if (n==0, 1, denominator(sumdiv(n, d, if (d%2, 1/d)))); \\ _Michel Marcus_, Jul 21 2020; corrected Jun 13 2022
%Y A098985 Cf. A000122, A098984, A098986.
%Y A098985 Cf. A336113 (numerators).
%K A098985 nonn,frac
%O A098985 0,4
%A A098985 _N. J. A. Sloane_, Oct 24 2004