OFFSET
1,2
COMMENTS
Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020
The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020
Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.
LINKS
Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixee de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
FORMULA
Equals 1/A005597.
Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....
EXAMPLE
Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577185338123...
MATHEMATICA
s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)
PROG
(PARI) prodeulerrat((1 + 1/(p*(p-2))), , 3) \\ Hugo Pfoertner, Aug 08 2020
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Jonathan Sondow, Nov 13 2009, Nov 17 2009
STATUS
approved