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A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1-1/p^2). 14
1, 0, 3, 4, 0, 1, 4, 8, 7, 5, 4, 1, 4, 3, 4, 1, 8, 8, 0, 5, 3, 9, 0, 3, 0, 6, 4, 4, 4, 1, 3, 0, 4, 7, 6, 2, 8, 5, 7, 8, 9, 6, 5, 4, 2, 8, 4, 8, 9, 0, 9, 9, 8, 8, 6, 4, 1, 6, 8, 2, 5, 0, 3, 8, 4, 2, 1, 2, 2, 2, 2, 4, 5, 8, 7, 1, 0, 9, 6, 3, 5, 8, 0, 4, 9, 6, 2, 1, 7, 0, 7, 9, 8, 2, 6, 2, 0, 5, 9, 6, 2, 8, 9, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Euler product of the Riemann zeta function at 2 restricted to primes in A002476, which is the inverse of the infinite product (1-1/7^2)*(1-1/13^2)*(1-1/19^2)*...

There is a complementary product_{primes == 2 mod 3} 1/(1-1/p^2) = A333240 = 1.4140643908921476375655018190798... such that (this constant here)*1.4140643.../(1-1/3^2) = zeta(2) = A013661.

Because 1/(1-p^(-2)) = 1+1/(p^2-1), the complementary 1.414064.. equals also Product_{primes p == 2 (mod 3)} (1+1/(p^2-1)), which appears in Eq. (1.8) of [Dence and Pomerance]. - R. J. Mathar, Jan 31 2013

LINKS

Table of n, a(n) for n=1..105.

Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, alternative link.

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26.

FORMULA

Equals 2*Pi^2 / (3^(7/2) * A301429^2). - Vaclav Kotesovec, May 12 2020

Equals Sum_{k>=1} 1/A004611(k)^2. - Amiram Eldar, Sep 27 2020

EXAMPLE

1.03401487541434188053903064441304762857896..

CROSSREFS

Cf. A002476, A004611, A007528, A175647, A334478, A334480.

Sequence in context: A021298 A170952 A194587 * A324362 A073234 A123685

Adjacent sequences:  A175643 A175644 A175645 * A175647 A175648 A175649

KEYWORD

cons,nonn,changed

AUTHOR

R. J. Mathar, Aug 01 2010

EXTENSIONS

More digits from Vaclav Kotesovec, May 12 2020 and Jun 27 2020

STATUS

approved

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Last modified October 1 00:54 EDT 2020. Contains 337440 sequences. (Running on oeis4.)