A395224 Numerator of (zeta(4*n) / zeta(2*n)^2) * (4^n + 1) / (4^n - 1) for n >= 1.

2, 34, 691, 1859138, 7159051, 56963745931, 383384156611, 1010491546156477058, 2868364599282829033657, 273765333712851131052871427, 2122567668414730590074148073, 94110177562207459848639528916696699, 15606769817411508637595836373988121, 5592915748426594283782949649676402543787

Offset

1,1

Comments

This sequence can be viewed as a variation of A114362/A114363, corresponding to the Euler product over odd primes p >= 3, i.e., Product_{p>=3} (p^(2n)-1)/(p^(2n)+1).

Links

Table of n, a(n) for n=1..14.

Formula

a(n) = numerator( ((2 * abs(bernoulli(4*n))*((2*n)!)^2)/(bernoulli(2*n)^2*(4*n)!))*(4^n + 1)/(4^n - 1) ).

a(n) = numerator( (zeta(4*n)/zeta(2*n)^2) * (2^(4*n) + 2^(2*n)) / (2^(4*n) - 2^(2*n)) ).

a(n) = numerator( Product_{p prime > 2} (p^(2*n) - 1) / (p^(2*n) + 1) ).

Example

For n = 2: (zeta(8)/zeta(4)^2) * (4^2 + 1)/(4^2 - 1) = ((Pi^8/9450) / (Pi^4/90)^2) * (17/15) = (6/7) * (17/15) = 34/35. The numerator is 34.
For n = 3: (zeta(12)/zeta(6)^2) * (4^3 + 1)/(4^3 - 1) = ((691*Pi^12/638512875) / (Pi^6/945)^2) * (65/63) = (691/715) * (65/63) = 691/693. The numerator is 691.

Prog

(PARI) {a(n) = numerator((2*abs(bernfrac(4*n))*(2*n)!^2) / (bernfrac(2*n)^2*(4*n)!) * (4^n+1)/(4^n-1))}

Crossrefs

Cf. A395225 (denominators), A000367, A002445 (Bernoulli numbers), A046988, A002432, A114362, A114363.

Sequence in context: A173208 A344107 A361773 * A224882 A227935 A384079

Adjacent sequences: A395220 A395221 A395222 * A395225 A395226 A395227

Keyword

nonn,frac

Author

Dimitris Valianatos, Apr 16 2026

Extensions

a(13) corrected by Sean A. Irvine, May 05 2026

a(14) from Dimitris Valianatos, May 06 2026

Status

approved