A088840 Denominator of sigma(4n)/sigma(n).

1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 127, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 31, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 21, 1, 1, 1, 7, 1, 1

Offset: 1

Author

Labos Elemer, Nov 04 2003

Keywords

• easy
• nonn
• mult
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sigma(n).

Crossrefs

See A088839 for numerator.

Cf. A088837, A088838, A088841, A038712, A080278, A000203, A001620, A007814, A193553, A213243.

Programs

Mathematica

Table[Denominator[DivisorSigma[1, 4*n]/DivisorSigma[1, n]], {n, 1, 128}]
a[n_] := Module[{e = IntegerExponent[n, 2]}, (((-1)^e+2)*(2^(e+1)-1))/3]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)

PARI

A088840(n) = denominator(sigma(4*n)/sigma(n)); \\ Antti Karttunen, Nov 18 2017

PARI

a(n) = {my(e = valuation(n, 2)); (((-1)^e+2) * (2^(e+1)-1))/3;} \\ Amiram Eldar, Oct 03 2023

Formula

From Amiram Eldar, Oct 03 2023: (Start)

Multiplicative with a(2^e) = (((-1)^e+2)*(2^(e+1)-1))/3 = A213243(e+1), and a(p^e) = 1 for an odd prime p.

a(n) = A213243(A007814(n+1)).

Dirichlet g.f.: ((8^s + 4^s + 2^(s+1))/(8^s + 4^s - 2^(s+2) - 4)) * zeta(s).

Sum_{k=1..n} a(k) = (2*n/(3*log(2))) * (log(n) + gamma - 1 + 7*log(2)/12), where gamma is Euler's constant (A001620). (End)

Extensions

Typo in definition corrected by Antti Karttunen, Nov 18 2017