A218342 Decimal expansion of e^-gamma * Product_(1 - 1/(p^3 - p^2 - p + 1)) where the product is over all primes p.

3, 4, 5, 3, 7, 2, 0, 6, 4, 1, 0, 2, 9, 8, 6, 4, 8, 7, 6, 7, 3, 4, 9, 6, 8, 2, 7, 8, 9, 1, 0, 3, 3, 7, 1, 0, 7, 2, 0, 6, 6, 5, 6, 2, 5, 3, 8, 0, 4, 1, 5, 8, 7, 2, 0, 5, 6, 0, 0, 4, 8, 9, 6, 6, 2, 5, 2, 6, 5, 3, 1, 9, 5, 0, 2, 2, 5, 1, 8, 6, 6, 9, 4, 7, 9, 0, 9, 1, 1, 6, 1, 3, 9, 2, 2, 7, 6, 3, 9, 6, 9, 6, 4, 4, 7

Offset: 0

Author

Charles R Greathouse IV, Oct 26 2012

Keywords

• nonn
• cons

Comments

The average order of Carmichael's lambda function is x/log x * exp(B log log x/log log log x (1 + o(1))), where B is this constant. Under the GRH, the same applies to A036391(n)/n, the sum of the orders mod n of the numbers coprime to n divided by n.

Examples

			0.34537206410298648767349682789103371072066562538041...
		

Links

• Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica 58 (1991), pp. 363-385.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C9).
• Sungjin Kim, On the order of 'a' modulo 'n' on average, International Journal of Number Theory, Vol. 12, No. 8 (2016), pp. 2073-2080; arXiv preprint, arXiv:1509.03768 [math.NT], 2015-2016.
• Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra & Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; arXiv preprint, arXiv:1108.5209 [math.NT], 2012.
• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011; Eq. (106) page 17.

Crossrefs

Cf. A002322, A036391, A080130, A162578.

Programs

Mathematica

$MaxExtraPrecision = 200; m0 = 1000; dm = 200; digits = 105; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - 1/((p - 1)^2*(p + 1))], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits + 10] &); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits + 5] !=  RealDigits[f[m - dm], 10, digits + 5], m = m + dm]; B = Exp[-EulerGamma]*f[m]; RealDigits[B, 10, digits] // First (* Jean-François Alcover, Sep 20 2015 *)

PARI

exp(-Euler) * prodeulerrat(1-1/((p-1)^2*(p+1))) \\ Amiram Eldar, Mar 09 2021

Extensions

More digits from Jean-François Alcover, Sep 20 2015