A275367 - OEIS

%I #44 Mar 20 2023 15:19:13

%S 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,

%T 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,

%U 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

%N Number of odd divisors of n^2.

%C All terms are odd.

%C First differs from A023136 at a(17).

%H Charles R Greathouse IV, <a href="/A275367/b275367.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A275367/a275367.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%F a(n) = A001227(n^2).

%F a(2n + 1) = A048691(2n + 1).

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

%F Multiplicative with a(2^e) = 1, a(p^e) = 2*e + 1 for odd prime p. - _Andrew Howroyd_, Jul 20 2018

%F Dirichlet g.f.: (zeta(s)^3/zeta(2*s))*(2^s-1)/(2^s+1). - _Amiram Eldar_, Dec 08 2022

%F 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

%p A275367 := proc(n) local a, d;

%p     a := 1 ;

%p     for d in ifactors(n)[2] do

%p         if op(1, d) > 2 then

%p             a := a*(2*op(2, d)+1) ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A275367(n),n=1..40) ; # _R. J. Mathar_, Mar 20 2023

%t Table[Count[Divisors[n^2], _?OddQ], {n, 120}] (* _Michael De Vlieger_, Jul 25 2016 *)

%t 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 *)

%o (PARI) a(n) = sumdiv(n^2, d, d%2); \\ _Michel Marcus_, Jul 25 2016

%o (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

%Y Cf. A000005, A001227, A023136, A048691.

%K nonn,easy,mult

%O 1,3

%A _Juri-Stepan Gerasimov_, Jul 24 2016