A324835 Decimal expansion of eta_4, a constant related to the asymptotic density of certain sets of residues.

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

Offset: 0

Author

Jean-François Alcover, Mar 17 2019

Keywords

• nonn
• cons

Examples

			0.0125930283986420136529911022622921769473432089854221861472578936695...
		

Links

• Carl Pomerance, Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory. 2011. Vol. 1. Iss. 1. pp. 52-66. See p. 62.

Crossrefs

Cf. A154945 (eta_1), A324833 (eta_2), A324834 (eta_3), A324836 (eta_5).

Programs

Mathematica

digits = 101; m0 = 2 digits; Clear[rd]; rd[m_] := rd[m] = RealDigits[eta4 = Sum[n(n+1)(n+2)/6 PrimeZetaP[2n+6], {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[eta4, digits]]; rd[m]

Formula

Sum_{p prime} 1/(p^2-1)^4.

Sum_{n>0} (n(n+1)(n+2)/6) P(2n+6) where P is the prime zeta P function.