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

0,2

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).

Links

Table of n, a(n) for n=0..104.

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)).

Example

0.13605447049622836522998926383768997616582469083783...

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]]

Prog

(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))

Crossrefs

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

Sequence in context: A141703 A283193 A021739 * A010470 A359295 A181916

Adjacent sequences: A382291 A382292 A382293 * A382295 A382296 A382297

Keyword

nonn,cons

Author

Amiram Eldar, Mar 21 2025

Status

approved