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

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, 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, 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

Offset: 0

Author

Artur Jasinski, Feb 09 2024

Keywords

• cons
• nonn

Comments

Definition:

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.

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

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

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

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

Limit_{k->oo} f(N*(k)) = 6/Pi^2 = 0.607927101854... see A059956.

Value computed and communicated by Bill Allombert.

Examples

			0.890925479476318332...
		

Links

• 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 right column).

Crossrefs

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

Programs

PARI

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, Feb 14 2014 [via Artur Jasinski]