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.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Oct 22 2014

Keywords

• nonn
• cons

Examples

			1.7426523110335154352489048069412986411544379898381...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 33.
• Carl Pomerance, The expected number of random elements to generate a finite abelian group, Periodica Mathematica Hungarica 43 (2001), 191-198.

Crossrefs

Cf. A021002, A033150, A084892, A084893, A249141.

Programs

Mathematica

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

PARI

default(realprecision,120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ Vaclav Kotesovec, Oct 22 2014

Formula

tau = sum_{j >= 1} (1-(1-2^(-j))*prod_{k >= j+1} zeta(k)^(-1)).

tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.