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A334912 a(n) = numerator (2^(4*n-1) * (2^(4*n-2) - 1) * (Bernoulli(4*n-2) / (4*n-2)!) * ((2*n-2)! / Euler(2*n-2))^2). 3
2, 16, 7936, 11184128, 209865342976, 2475749026562048, 123460740095103991808, 5779796046952399460368384, 3729407703720529571097509625856, 485491405392529556189699853976076288, 193817991886041515914007312001087567822848, 56920344782482721622150071084079041150980194304 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For every s odd power of odd prime number p applies: pi(p; 4, 3) = pi(p^s; 4, 3) and pi(p; 4, 1) = pi(p^s; 4, 1).

Product_{p = A002144} ((p^(2*n - 1) - 1) / (p^(2*n - 1) + 1)) = (2^(2*n) + 2) * (2*n - 2)! * (Pi^(2*n - 1) / zeta(2*n - 1)) * (zeta(4*n - 2) / Pi^(4*n - 2)) / abs(EulerE(2*n - 2)), n > 1.

Product_{p = A002145} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1)) = (2^(2*n) - 2) * (2*n - 2)! * (zeta(2*n - 1) / Pi^(2*n - 1)) / abs(EulerE(2*n - 2)), n > 1.

Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = (2^(4*n - 1) * (2^(4*n - 2) - 1) * (BernoulliB(4*n - 2) / (4*n - 2)!) * ((2*n - 2)! / EulerE(2*n - 2))^2 ) = a(n) / A334835(n).

LINKS

Table of n, a(n) for n=1..12.

X. Gourdon and P. Sebah, Some Constants from Number theory

FORMULA

a(n) = numerator (Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = numerator (2^(4*n) - 4) * ((2*n - 2)! / EulerE(2*n - 2))^2 * (zeta(4*n - 2) / Pi^(4*n - 2))).

From Vaclav Kotesovec, May 17 2020: (Start)

a(n) / A334835(n) ~ 1 as n tends to infinity.

a(n) = numerator((1 - 1/2^(4*n-2)) * zeta(4*n-2) / DirichletBeta(2*n-1)^2). (End)

MATHEMATICA

Numerator[Table[2^(4*s - 1) * (2^(4*s - 2) - 1) * BernoulliB[4*s - 2] * (2*s - 2)!^2 / (EulerE[2*s - 2]^2 * (4*s - 2)!), {s, 1, 15}]] (* or *) Numerator[Table[(1 - 1/2^(4*s - 2))*Zeta[4*s - 2]/DirichletBeta[2*s - 1]^2, {s, 1, 15}]] (* Vaclav Kotesovec, May 17 2020 *)

PROG

(PARI) E(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1); \\ see A000364

a(n) = numerator((2^(4*n-1)*(2^(4*n-2)-1)*(bernfrac(4*n-2)/(4*n-2)!)*((2*n-2)!/ E(n-1))^2)); \\ Michel Marcus, May 17 2020

CROSSREFS

Cf. A000040, A065091, A002144, A002145, A334835 (denominators).

Cf. A000364, A027641/A027642.

Sequence in context: A324565 A306729 A325049 * A092798 A333540 A258169

Adjacent sequences:  A334909 A334910 A334911 * A334913 A334914 A334915

KEYWORD

nonn,frac

AUTHOR

Dimitris Valianatos, May 16 2020

EXTENSIONS

More terms from Michel Marcus, May 17 2020

STATUS

approved

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Last modified August 4 18:59 EDT 2021. Contains 346455 sequences. (Running on oeis4.)