A118261 Decimal expansion of probability of a weakly carefree couple.

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

Offset

0,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.

Links

Table of n, a(n) for n=0..99.

Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.

Eric Weisstein's World of Mathematics, Carefree Couple.

Formula

Equals 2*K1 - K2, where K1 = A065464 and K2 = A065473.

Example

0.5697515829197101463296387...

Mathematica

$MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
K1 = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
K2 = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
RealDigits[2 K1 - K2, 10, digits][[1]] (* Jean-François Alcover, May 15 2016 *)

Prog

(PARI) 2 * prodeulerrat(1 - (2*p-1)/p^3) - prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 03 2021

Crossrefs

Cf. A065464, A065473, A118259, A118260.

Sequence in context: A340565 A197283 A244843 * A246749 A357471 A021641

Adjacent sequences: A118258 A118259 A118260 * A118262 A118263 A118264

Keyword

nonn,cons

Author

Eric W. Weisstein, Apr 20 2006

Status

approved