A275367 Number of odd divisors of n^2.

1, 1, 3, 1, 3, 3, 3, 1, 5, 3, 3, 3, 3, 3, 9, 1, 3, 5, 3, 3, 9, 3, 3, 3, 5, 3, 7, 3, 3, 9, 3, 1, 9, 3, 9, 5, 3, 3, 9, 3, 3, 9, 3, 3, 15, 3, 3, 3, 5, 5, 9, 3, 3, 7, 9, 3, 9, 3, 3, 9, 3, 3, 15, 1, 9, 9, 3, 3, 9, 9, 3, 5, 3, 3, 15, 3, 9, 9, 3, 3, 9, 3, 3, 9, 9, 3, 9, 3, 3, 15, 9, 3, 9

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).

a(n) = A000005(n^2) if n is odd, else A000005(2*n^2) - A000005(n^2).

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

Cf. A000005, A001227, A023136, A048691.

Sequence in context: A072219 A173854 A059789 * A023136 A337333 A348665

Adjacent sequences: A275364 A275365 A275366 * A275368 A275369 A275370

Keyword

nonn,easy,mult

Author

Juri-Stepan Gerasimov, Jul 24 2016

Status

approved