A084945 Decimal expansion of Golomb-Dickman constant.

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

Offset: 0

Author

Eric W. Weisstein, Jun 13 2003

Keywords

• nonn
• cons

Comments

The first 27 digits form a prime. - Jonathan Vos Post, Nov 12 2004

The first 1659 digits form a prime. - David Broadhurst, Apr 02 2010

The average number of digits in the largest prime factor of a random x-digit number is asymptotically x times this constant. - Charles R Greathouse IV, Jul 28 2015

Named after the American mathematician Solomon W. Golomb (1932 - 2016) and the Swedish actuary Karl Dickman (1861 - 1947). - Amiram Eldar, Aug 25 2020

Examples

			0.62432998854355087...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 284-287.

Links

• David Broadhurst, PrimeForm message on the first 1659 digits, Apr 02 2010.
• David Broadhurst, Titanic Golomb-Dickman prime, digest of 5 messages in primeform Yahoo group, Apr 2 - Apr 9, 2010. [Cached copy]
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.
• Solomon W. Golomb, Research Problem 11: Random permutations, Bull. Amer. Math. Soc. 70 (1964), p. 747.
• Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, preprint, arXiv:1303.1856 [math.NT], 2013.
• Andrew MacFie and Daniel Panario, Random Mappings with Restricted Preimages, in Progress in Cryptology-LATINCRYPT 2012, LNCS 7533, pp. 254-270, 2012. - From _N. J. A. Sloane_, Dec 25 2012
• Simon Plouffe, The Golomb constant.
• Eric Weisstein's World of Mathematics, Golomb-Dickman Constant.
• Wikipedia, Golomb-Dickman constant.

Crossrefs

Cf. A174974, A174975, A225336.

Programs

Maple

E1:= z-> int(exp(-t)/t, t=z..infinity):
lambda:= int(exp(-x-E1(x)), x=0..infinity):
s:= convert(evalf(lambda, 130), string):
seq(parse(s[n+1]), n=1..120); # Alois P. Heinz, Nov 20 2011

Mathematica

NIntegrate[Exp[LogIntegral[x]], {x, 0, 1}, WorkingPrecision->110, MaxRecursion->20]

PARI

intnum(x=0,1-1e-9,exp(-eint1(-log(x)))) \\ Charles R Greathouse IV, Jul 28 2015

PARI

default(realprecision, 103);
limitnum(n->intnum(x=0, 1-1/n, exp(-eint1(-log(x))))) \\ Gheorghe Coserea, Sep 26 2018

Formula

From Amiram Eldar, Aug 25 2020: (Start)

Equals Integral_{x=0..1} exp(li(x)) dx, where li(x) is the logarithmic integral.

Equals Integral_{x=0..oo} exp(-x + Ei(-x)) dx, where Ei(x) is the exponential integral.

Equals Integral_{x=0..1} F(x/(1-x)) dx, where F(x) is the Dickman function. (End)