A380602 Decimal expansion of the asymptotic mean of the ratio A000005(k)/A322483(k).

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

Offset: 1

Author

Amiram Eldar, Jan 27 2025

Keywords

• nonn
• cons

Examples

			1.23587977752617354871093805318945110447752750370305...
		

Crossrefs

Cf. A000005, A322483, A380601 (mean of the inverse ratio).

Similar constants: A307869, A361059, A361061.

Programs

Mathematica

$MaxExtraPrecision = 1000; m = 1000; f[x_] := 2/x - (1 + 1/x - 2/x^2)*Log[1-x^2]/x; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

PARI

default(realprecision, 120); default(parisize, 30000000);
my(m = 1024, x = 'x + O('x^m), v); v = Vec(2/x - (1 + 1/x - 2/x^2)*log(1-x^2)/x); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A000005(k)/A322483(k).

Equals Product_{p prime} (2*p - (1+p-2*p^2)*log(1-1/p^2)*p).