A372765 - OEIS

%I #54 Jun 03 2024 18:17:39

%S 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,

%T 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,

%U 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

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

%C Definition:

%C 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.

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

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

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

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

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

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

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

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

%C Value computed and communicated by Bill Allombert.

%H Bill Allombert, <a href="/A372765/a372765.txt">Results of pari computation of Lichtman constants f(N(k)) with precision 500 decimals for k=1..20, email 20.06.2023</a>

%H Jared Duker Lichtman, <a href="https://arxiv.org/abs/1909.00804">Almost primes and the Banks-Martin conjecture</a>, arXiv:1909.00804 [math.NT], 2019 (Figure 2 left column).

%e 1.1448165734059179915...

%o (PARI) pz(x)= sum(n=1, max(2, bitprecision(x)/x), my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));

%o Lichtman(n)=intnum(s=1, [oo, log(2)], exp(sum(i=1, n, pz(i*s)*x^i/i)+O(x^(n+1)))-1)

%o Lichtman(20)

%o \\ Bill Allombert, May 14 2024 [via Artur Jasinski]

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

%K nonn,cons

%O 1,3

%A _Artur Jasinski_, May 14 2024