cp's OEIS Frontend

A275367 Number of odd divisors of n^2.

%I A275367 #44 Mar 20 2023 15:19:13
%S A275367 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 A275367 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 A275367 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 A275367 Number of odd divisors of n^2.
%C A275367 All terms are odd.
%C A275367 First differs from A023136 at a(17).
%H A275367 Charles R Greathouse IV, <a href="/A275367/b275367.txt">Table of n, a(n) for n = 1..10000</a>
%H A275367 Vaclav Kotesovec, <a href="/A275367/a275367.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A275367 a(n) = A001227(n^2).
%F A275367 a(2n + 1) = A048691(2n + 1).
%F A275367 a(n) = A000005(n^2) if n is odd, else A000005(2*n^2) - A000005(n^2).
%F A275367 Multiplicative with a(2^e) = 1, a(p^e) = 2*e + 1 for odd prime p. - _Andrew Howroyd_, Jul 20 2018
%F A275367 Dirichlet g.f.: (zeta(s)^3/zeta(2*s))*(2^s-1)/(2^s+1). - _Amiram Eldar_, Dec 08 2022
%F A275367 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 A275367 := proc(n) local a, d;
%p A275367     a := 1 ;
%p A275367     for d in ifactors(n)[2] do
%p A275367         if op(1, d) > 2 then
%p A275367             a := a*(2*op(2, d)+1) ;
%p A275367         end if;
%p A275367     end do:
%p A275367     a ;
%p A275367 end proc:
%p A275367 seq(A275367(n),n=1..40) ; # _R. J. Mathar_, Mar 20 2023
%t A275367 Table[Count[Divisors[n^2], _?OddQ], {n, 120}] (* _Michael De Vlieger_, Jul 25 2016 *)
%t A275367 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 A275367 (PARI) a(n) = sumdiv(n^2, d, d%2); \\ _Michel Marcus_, Jul 25 2016
%o A275367 (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 A275367 Cf. A000005, A001227, A023136, A048691.
%K A275367 nonn,easy,mult
%O A275367 1,3
%A A275367 _Juri-Stepan Gerasimov_, Jul 24 2016