cp's OEIS Frontend

A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.

%I A245055 #16 Jan 17 2020 05:45:03
%S A245055 1,7,4,2,6,5,2,3,1,1,0,3,3,5,1,5,4,3,5,2,4,8,9,0,4,8,0,6,9,4,1,2,9,8,
%T A245055 6,4,1,1,5,4,4,3,7,9,8,9,8,3,8,1,0,4,6,2,8,1,4,2,9,0,4,7,9,5,7,4,6,5,
%U A245055 5,5,0,3,8,7,0,0,8,1,3,5,0,8,6,8,0,5,8,1,4,7,4,1,7,5,2,4,7,8,8,1,2
%N A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.
%D A245055 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.
%H A245055 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 33.
%H A245055 Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/PDF/paper123.pdf">The expected number of random elements to generate a finite abelian group</a>, Periodica Mathematica Hungarica 43 (2001), 191-198.
%F A245055 tau = sum_{j >= 1} (1-(1-2^(-j))*prod_{k >= j+1} zeta(k)^(-1)).
%F A245055 tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.
%e A245055 1.7426523110335154352489048069412986411544379898381...
%t A245055 digits = 101; max = 400; c = 1/Product[N[Zeta[k], digits + 100], {k, 2, max}]; p[j_] := Product[N[Zeta[k], digits + 100], {k, 2, j}]; tau = Sum[1 - (1 - 2^-j)*c*p[j], {j, 1, max}]; RealDigits[tau, 10, digits ] // First
%o A245055 (PARI) default(realprecision,120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ _Vaclav Kotesovec_, Oct 22 2014
%Y A245055 Cf. A021002, A033150, A084892, A084893, A249141.
%K A245055 nonn,cons
%O A245055 1,2
%A A245055 _Jean-François Alcover_, Oct 22 2014