A098985 Denominators in series expansion of log( Sum_{m=-oo,oo} q^(m^2) ).
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, 29, 5, 31, 1, 11, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 49, 25, 17, 13, 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
Offset: 0
Examples
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-...
Links
- Sumit Kumar Jha, 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, arXiv:2007.04243 [math.GM], 2020.
Programs
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Maple
A098985_list := proc(n::integer) local q,m,nsq ; nsq := floor(sqrt(n)) ; add(q^(m^2),m=-nsq-1..nsq+1) ; log(%) ; taylor(%,q=0,n+1) ; [seq( denom(coeftayl(%,q=0,i)) ,i=1..n) ] ; end proc: A098985_list(200) ; # R. J. Mathar, Jul 16 2020 A336114 := proc(n::integer) local a ; for d in numtheory[divisors](n) do if type(d,'odd') then a := a+1/d ; end if; end do; denom(a) ; end proc: seq(A336114(n),n=1..70) ; # R. J. Mathar, Jul 16 2020
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Mathematica
Denominator[CoefficientList[Series[Log[Sum[q^m^2, {m, -Infinity, Infinity}]], {q, 0, 79}], q]] (* L. Edson Jeffery, Jul 14 2014 *) a[n_] := Denominator @ DivisorSum[n, 1/# &, OddQ[#] &]; Array[a, 100] (* Amiram Eldar, Jul 09 2020 *) -
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 -
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
Comments