A382294 Decimal expansion of the asymptotic mean of the excess of the number of Fermi-Dirac factors of k over the number of distinct prime factors of k when k runs over the positive integers.

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

Offset: 0

Author

Amiram Eldar, Mar 21 2025

Keywords

• nonn
• cons

Comments

Analogous to Sum_{p prime} 1/(p*(p-1)) (A136141), which is the asymptotic mean of the excess of the number of prime factors over the number of distinct prime factors (A046660).

Examples

			0.13605447049622836522998926383768997616582469083783...
		

Crossrefs

Cf. A001221, A046660, A064547, A088705, A136141, A382290.

Programs

Mathematica

s[n_] := Module[{c = CoefficientList[Series[-x + Sum[x^(2^k)/(1+x^(2^k)), {k, 0, n}],{x, 0, 2^n}], x]},Sum[c[[i]] * PrimeZetaP[i-1], {i, 3, Length[c]-2}]]; RealDigits[s[10], 10, 120][[1]]

PARI

default(realprecision, 120); default(parisize, 10000000);
f(x, n) = -x + sum(k = 0, n, x^(2^k)/(1+x^(2^k)));
sumeulerrat(f(1/p, 8))

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A382290(k).

Equal Sum_{k>=3} A088705(k) * P(k), where P(s) is the prime zeta function.

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