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Decimal expansion of the probability that an integer 4-tuple is pairwise coprime.
2

%I #22 Jan 05 2025 19:51:41

%S 1,1,4,8,8,4,0,4,4,0,8,0,2,2,8,7,8,8,7,2,9,2,5,1,2,7,6,7,0,1,5,9,9,0,

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

%U 6,6,7,2,8,0,1,6,1,6,7,4,6,1,7,4,0,2,3

%N Decimal expansion of the probability that an integer 4-tuple is pairwise coprime.

%H László Tóth, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-1/toth.pdf">The probability that k positive integers are pairwise relatively prime</a>, Fibonacci Quarterly, Vol. 40, No. 1 (2002), pp. 13-18.

%F Equals Product_{p prime} (1 - 1/p)^3 * (1 + 3/p).

%e 0.114884044080228788729251276701599097848713552687283...

%t $MaxExtraPrecision = 1000; nm = 1000; c = LinearRecurrence[{-2, 3}, {0, -12}, nm]; f[x_] := (1 - x)^3*(1 + 3*x); RealDigits[f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

%o (PARI) prodeulerrat((1 - 1/p)^3 * (1 + 3/p)) \\ _Amiram Eldar_, Jun 29 2023

%Y Cf. A059956, A065473, A256392.

%K nonn,cons

%O 0,3

%A _Amiram Eldar_, Aug 27 2019