OFFSET
1,3
COMMENTS
All terms are odd.
First differs from A023136 at a(17).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
FORMULA
a(n) = A001227(n^2).
a(2n + 1) = A048691(2n + 1).
Multiplicative with a(2^e) = 1, a(p^e) = 2*e + 1 for odd prime p. - Andrew Howroyd, Jul 20 2018
Dirichlet g.f.: (zeta(s)^3/zeta(2*s))*(2^s-1)/(2^s+1). - Amiram Eldar, Dec 08 2022
Sum_{k=1..n} a(k) ~ n*log(n)^2/Pi^2 + 2*n*log(n)*((3*gamma + 4*log(2)/3 - 1)/Pi^2 - 12*zeta'(2)/Pi^4) + 2*n*((1 + 3*gamma^2 - 4*log(2)/3 - 2*log(2)^2/9 + gamma*(4*log(2) - 3) - 3*sg1)/Pi^2 - 4*((9*gamma*zeta'(2) + (4*log(2) - 3)*zeta'(2) + 3*zeta''(2))/Pi^4) + 144*zeta'(2)^2/Pi^6), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Dec 08 2022
MAPLE
A275367 := proc(n) local a, d;
a := 1 ;
for d in ifactors(n)[2] do
if op(1, d) > 2 then
a := a*(2*op(2, d)+1) ;
end if;
end do:
a ;
end proc:
seq(A275367(n), n=1..40) ; # R. J. Mathar, Mar 20 2023
MATHEMATICA
Table[Count[Divisors[n^2], _?OddQ], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)
f[2, e_] := 1; f[p_, e_] := 2*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)
PROG
(PARI) a(n) = sumdiv(n^2, d, d%2); \\ Michel Marcus, Jul 25 2016
(PARI) a(n)=my(f=factor(n>>valuation(n, 2))[, 2]); prod(i=1, #f, 2*f[i]+1) \\ Charles R Greathouse IV, Jul 28 2016
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Juri-Stepan Gerasimov, Jul 24 2016
STATUS
approved