cp's OEIS Frontend

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

%I A372765 #57 Mar 30 2025 02:40:06
%S A372765 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 A372765 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 A372765 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 A372765 Decimal expansion of Lichtman constant f(N(2)).
%C A372765 Definition:
%C A372765 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 A372765 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 A372765 Lichtman constant f(N(1)) see A137245.
%C A372765 Lichtman constant f(N(2)) this sequence.
%C A372765 Lichtman constant f(N(3)) see A372827.
%C A372765 Lichtman constant f(N(4)) see A372828.
%C A372765 Minimal value of f(N(k)) occurs for k=6 f(N(6)) = 0.9887534530145...
%C A372765 For k>=6, 1 > f(N(k+1)) > f(N(k)).
%C A372765 When k -> oo then f(N(k)) -> 1.
%C A372765 Value computed and communicated by Bill Allombert.
%H A372765 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 A372765 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 A372765 1.1448165734059179915...
%o A372765 (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 A372765 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 A372765 Lichtman(20)
%o A372765 \\ Bill Allombert, May 14 2024 [via Artur Jasinski]
%Y A372765 Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372827, A372828.
%K A372765 nonn,cons
%O A372765 1,3
%A A372765 _Artur Jasinski_, May 14 2024