OFFSET
0,1
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
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.
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)
EXAMPLE
0.62432998854355087...
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]
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jun 13 2003
STATUS
approved