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

0,4

Comments

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

Links

Table of n, a(n) for n=0..79.

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.

Example

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-...

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

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 *)

Prog

(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

Crossrefs

Cf. A000122, A098984, A098986.

Cf. A336113 (numerators).

Sequence in context: A336650 A343249 A327656 * A356168 A327539 A072963

Adjacent sequences: A098982 A098983 A098984 * A098986 A098987 A098988

Keyword

nonn,frac

Author

N. J. A. Sloane, Oct 24 2004

Status

approved