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%I #4 Mar 17 2019 21:11:46
%S 4,1,4,5,8,7,3,4,7,5,2,5,8,9,5,6,1,9,5,3,2,4,6,7,8,8,7,0,5,9,0,5,1,7,
%T 8,1,7,5,4,1,9,3,1,4,0,5,6,2,2,8,7,9,2,9,9,5,5,7,8,8,1,3,1,2,8,0,6,4,
%U 9,6,5,3,8,8,4,7,0,0,9,6,9,7,9,1,4,6,8,1,6,9,9,7,8,5,5,0,3,5,1,4,7,9
%N Decimal expansion of eta_5, a constant related to the asymptotic density of certain sets of residues.
%H Carl Pomerance, Andrzej Schinzel, <a href="http://mjcnt.phystech.edu/en/article.php?id=4">Multiplicative Properties of Sets of Residues</a>, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.
%F Sum_{p prime} 1/(p^2-1)^5.
%F Sum_{n>0} (n(n-1)(n-2)(n-3)/24) P(2n+2) where P is the prime zeta P function.
%e 0.0041458734752589561953246788705905178175419314056228792995578813128...
%t digits = 102; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta5 = Sum[n(n-1)(n-2)(n-3)/24 PrimeZetaP[2n+2], {n, 1, m}], 10, digits][[1]]; rd[m0]; rd[m = 2 m0]; While[rd[m] != rd[m - m0], Print[m]; m = m + m0]; Print[N[eta5, digits]]; rd[m]
%Y Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324835 (eta_4).
%K nonn,cons
%O -2,1
%A _Jean-François Alcover_, Mar 17 2019
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