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

1,2

References

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

Links

Table of n, a(n) for n=1..101.

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.

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.

Example

1.7426523110335154352489048069412986411544379898381...

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

Prog

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

Crossrefs

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

Sequence in context: A198356 A254349 A019608 * A335020 A225410 A248750

Adjacent sequences: A245052 A245053 A245054 * A245056 A245057 A245058

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 22 2014

Status

approved