A395225 Denominator of (zeta(4*n) / zeta(2*n)^2) * (4^n + 1)/(4^n - 1) for n >= 1.

3, 35, 693, 1859715, 7159295, 56963960781, 383384317050, 1010491593118379235, 2868364614091837239225, 273765333869887141063556775, 2122567668550009363293643890, 94110177562873895978541410756672925, 15606769817423788433224704002835378, 5592915748427083243867217876905465272970

Offset

1,1

Comments

This sequence is the denominator analog of A395224 and can be viewed as a variation of A114362/A114363, removing the contribution of the prime p=2 from the Euler product.

Links

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

Formula

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

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

a(n) = denominator(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) = 102/105 = 34/35. The denominator is 35.
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 denominator is 693.

Prog

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

Crossrefs

Cf. A395224 (numerators), A000367, A002445 (Bernoulli numbers), A046988, A002432, A114362, A114363.

Sequence in context: A304191 A009071 A093531 * A129505 A185752 A379659

Adjacent sequences: A395221 A395222 A395224 * A395226 A395227 A395228

Keyword

nonn,frac

Author

Dimitris Valianatos, Apr 16 2026

Extensions

Incorrect a(14)-a(15) removed by Sean A. Irvine, May 05 2026

a(14) from Dimitris Valianatos, May 06 2026

Status

approved