A370093 - OEIS

%I #37 Mar 09 2024 11:29:37

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

%C Definition:

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

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

%C Lichtman constant f(N*(3)) see A370112.

%C Lichtman constant f(N*(4)) see A370113.

%C when k -> oo than f(N*(k) -> 6/Pi^2 = 0.607927101854... see A059956.

%C Value computed and communicated by Bill Allombert.

%H 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 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 0.890925479476318332...

%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, Feb 14 2014 [via Artur Jasinski]

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

%K cons,nonn

%O 0,1

%A _Artur Jasinski_, Feb 09 2024