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

0,1

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

Table of n, a(n) for n=0..101.

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

Cf. A174974, A174975, A225336.

Sequence in context: A354782 A064925 A173273 * A370465 A358089 A010493

Adjacent sequences: A084942 A084943 A084944 * A084946 A084947 A084948

Keyword

nonn,cons

Author

Eric W. Weisstein, Jun 13 2003

Status

approved