cp's OEIS Frontend

A370093 Decimal expansion of Lichtman constant f(N*(2)).

%I A370093 #43 Apr 13 2025 07:23:52
%S A370093 8,9,0,9,2,5,4,7,9,4,7,6,3,1,8,3,3,2,1,3,7,2,6,2,6,2,1,9,9,5,9,8,8,2,
%T A370093 9,3,8,9,7,8,1,8,1,3,8,1,6,5,2,7,6,3,8,9,8,3,2,9,0,7,5,6,6,9,9,8,9,1,
%U A370093 3,4,4,1,0,6,1,4,5,0,5,2,0,7,3,6,6,4,9,7,3,3,5,9,2,7,6,2,3,2,7,5,0,3,3,3,8,3
%N A370093 Decimal expansion of Lichtman constant f(N*(2)).
%C A370093 Definition:
%C A370093 f(N*(k)) = Integral_{s>=1} P_k*(s), where P_k*(s) = Sum_{n>1 and (big) Omega(n)=k} mu(n)^2/n^s, where mu is Möbius (or Moebius) Mu function see A008683, and (big) Omega is number of prime divisors of n counted with multiplicity see A001222.
%C A370093 Lichtman constant f(N*(1)) see A137245.
%C A370093 Lichtman constant f(N*(2)) this sequence.
%C A370093 Lichtman constant f(N*(3)) see A370112.
%C A370093 Lichtman constant f(N*(4)) see A370113.
%C A370093 Limit_{k->oo} f(N*(k)) = 6/Pi^2 = 0.607927101854... see A059956.
%C A370093 Value computed and communicated by Bill Allombert.
%H A370093 Bill Allombert, <a href="/A370093/a370093.txt">Results of pari computation of Lichtman constants f(N*(k)) with precision 500 decimals for k=1..20</a>, email 20.06.2023.
%H A370093 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 right column).
%e A370093 0.890925479476318332...
%o A370093 (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 A370093 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 A370093 Lichtman(20)
%o A370093 \\ Bill Allombert, Feb 14 2014 [via Artur Jasinski]
%Y A370093 Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370112, A370113.
%K A370093 cons,nonn
%O A370093 0,1
%A A370093 _Artur Jasinski_, Feb 09 2024