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
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Programs
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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
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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 *)
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PARI
a(n) = sumdiv(n^2, d, d%2); \\ Michel Marcus, Jul 25 2016
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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
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
Comments