A231327 -id:A231327 - cp's OEIS frontend

A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

1, 1, 1, 691, 3617, 174611, 236364091, 3392780147, 7709321041217, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 5609403368997817686249127547, 61628132164268458257532691681, 354198989901889536240773677094747

Offset: 0

Leo Depuydt, Nov 07 2013

Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).
For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.
The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - Leo Depuydt, Nov 22 2013, Nov 27 2013
The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - Leo Depuydt, Aug 22 2014
From Robert Israel, Aug 22 2014: (Start)
Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).
Not the same as abs(A001067(2*n)): they differ first at n=17.
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, 1960, p. 255.

Leo Depuydt, Details, including proof and relation to zeta function

Cf. A231327 (corresponding denominator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, A156036 (same number sequence, though in various transformations (alternation of signs, intervening numbers, and so on)).
Cf. A027641 and A027642

seq(numer((-1)^n*bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Numerator[Table[Zeta[4n]/(Zeta[2n] * Pi^(2n)), {n, 0, 15}]] (* T. D. Noe, Nov 18 2013 *)

A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4096, 4225, 4356

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Differs from its subsequence A340674 by having the terms 64, 729, 1024, 4096, .... .

Numbers whose prime factorization exponents are equal and even.

Intersection of A000290 and A072777.
Equals A072777 \ A384518.
A340674 is a subsequence.
Cf. A005117, A062770, A072774, A231273, A231327.

Select[Range[2, 100], SameQ @@ FactorInteger[#][[;;, 2]] &]^2

isok(k) = {my(s, e = ispower(k, , &s)); !(e % 2) && issquarefree(s);}

from math import isqrt
from sympy import mobius, integer_nthroot
def A384517(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1		
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(2,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

a(n) = A062770(n)^2 = A072774(n+1)^2.

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k)/zeta(4*k)-1) = Sum{k>=1} (A231327(k)/(A231273(k)*Pi^(2*k)) - 1) = 0.62022193512079649421... .

cp's OEIS Frontend

A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).

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