A324834 - OEIS

%I #4 Mar 17 2019 21:11:29

%S 3,9,0,7,2,4,0,5,7,3,5,5,7,4,7,9,1,8,8,7,6,5,9,5,0,3,3,2,0,4,2,2,9,7,

%T 6,3,8,6,6,8,4,8,3,8,2,4,4,7,7,3,3,6,0,3,5,6,7,5,4,0,6,6,0,3,2,6,9,1,

%U 7,5,8,3,7,6,1,9,2,4,9,2,0,2,9,8,1,7,9,1,0,0,6,9,0,7,6,8,0,0,5,6,2,3

%N Decimal expansion of eta_3, 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)^3.

%F Sum_{n>0} (n(n+1)/2) P(2n+4) where P is the prime zeta P function.

%e 0.03907240573557479188765950332042297638668483824477336035675406603269...

%t digits = 102; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta3 = Sum[n (n+1)/2 PrimeZetaP[2 n + 4], {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[eta3, digits]]; rd[m]

%Y Cf. A154945 (eta_1), A324833 (eta_2), A324835 (eta_4), A324836 (eta_5).

%K nonn,cons

%O -1,1

%A _Jean-François Alcover_, Mar 17 2019