A372765 Decimal expansion of Lichtman constant f(N(2)).

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

Offset

1,3

Comments

Definition:

f(N(k)) = Sum_{n>1 and (big) Omega(n)=k} 1/(n*log(n)), where (big) Omega is number of prime divisors of n counted with multiplicity see A001222.

f(N(k)) = Integral_{s>=1} P_k(s), where P_k(s) = Sum_{n>1 and (big) Omega(n)=k} 1/n^s.

Lichtman constant f(N(1)) see A137245.

Lichtman constant f(N(2)) this sequence.

Lichtman constant f(N(3)) see A372827.

Lichtman constant f(N(4)) see A372828.

Minimal value of f(N(k)) occurs for k=6 f(N(6)) = 0.9887534530145...

For k>=6 1>f(N(k+1))>f(N(k).

When k -> oo then f(N(k)) -> 1.

Value computed and communicated by Bill Allombert.

Links

Table of n, a(n) for n=1..105.

Bill Allombert, Results of pari computation of Lichtman constants f(N(k)) with precision 500 decimals for k=1..20, email 20.06.2023

Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019 (Figure 2 left column).

Example

1.1448165734059179915...

Prog

(PARI) pz(x)= sum(n=1, max(2, bitprecision(x)/x), my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
Lichtman(n)=intnum(s=1, [oo, log(2)], exp(sum(i=1, n, pz(i*s)*x^i/i)+O(x^(n+1)))-1)
Lichtman(20)
\\ Bill Allombert, May 14 2024 [via Artur Jasinski]

Crossrefs

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372827, A372828.

Sequence in context: A273118 A273690 A010660 * A067736 A181387 A091671

Adjacent sequences: A372762 A372763 A372764 * A372766 A372767 A372768

Keyword

nonn,cons

Author

Artur Jasinski, May 14 2024

Status

approved