A372828 -id:A372828 - cp's OEIS frontend

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

Artur Jasinski, May 14 2024

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.

			1.1448165734059179915...

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).

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

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]

A372827 Decimal expansion of Lichtman constant f(N(3)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, May 14 2024

For definition and links see A372765.

			1.0308351017932175719...

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372765, A372828.

cp's OEIS Frontend

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

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