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A118261 Decimal expansion of probability of a weakly carefree couple. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A021641 A332327

Adjacent sequences:  A118258 A118259 A118260 * A118262 A118263 A118264

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Apr 20 2006

STATUS

approved

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Last modified October 27 02:00 EDT 2021. Contains 348270 sequences. (Running on oeis4.)