A382295 Decimal expansion of the asymptotic mean of the number of ways to factor k into "Fermi-Dirac primes" when k runs over the positive integers.

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

Offset: 1

Author

Amiram Eldar, Mar 21 2025

Keywords

• nonn
• cons

Examples

			1.78763680016944566698863293948945988146590046137002...
		

Crossrefs

Cf. A005117 (positions of 1's in A050377), A050377, A082293 (positions of 2's), A330687 (positions of records).

Programs

Mathematica

$MaxExtraPrecision = 1500; m = 1500; em = 50; f[x_] := Log[1-x] - Sum[Log[1-x^(2^k)], {k, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

PARI

default(realprecision, 120); default(parisize, 10000000);
f(x, n) = (1-x) / prod(k = 0, n, (1 - x^(2^k)));
prodeulerrat(f(1/p, 10))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A050377(k).

Equals Product_{p prime} f(1/p), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)).