A084945 - OEIS

%I #64 Aug 15 2023 12:07:13

%S 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,

%T 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,

%U 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

%N Decimal expansion of Golomb-Dickman constant.

%C The first 27 digits form a prime. - _Jonathan Vos Post_, Nov 12 2004

%C The first 1659 digits form a prime. - _David Broadhurst_, Apr 02 2010

%C 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

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

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

%H David Broadhurst, <a href="http://groups.yahoo.com/group/primeform/message/10249?v=0">PrimeForm message</a> on the first 1659 digits, Apr 02 2010.

%H David Broadhurst, <a href="/A084945/a084945.txt">Titanic Golomb-Dickman prime</a>, digest of 5 messages in primeform Yahoo group, Apr 2 - Apr 9, 2010. [Cached copy]

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.

%H Solomon W. Golomb, <a href="http://dx.doi.org/10.1090/S0002-9904-1964-11224-6">Research Problem 11: Random permutations</a>, Bull. Amer. Math. Soc. 70 (1964), p. 747.

%H Jeffrey C. Lagarias, <a href="https://doi.org/10.1090/S0273-0979-2013-01423-X">Euler's constant: Euler's work and modern developments</a>, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, <a href="http://arxiv.org/abs/1303.1856">preprint</a>, arXiv:1303.1856 [math.NT], 2013.

%H Andrew MacFie and Daniel Panario, <a href="http://dx.doi.org/10.1007/978-3-642-33481-8_14">Random Mappings with Restricted Preimages</a>, in Progress in Cryptology-LATINCRYPT 2012, LNCS 7533, pp. 254-270, 2012. - From _N. J. A. Sloane_, Dec 25 2012

%H Simon Plouffe, <a href="http://plouffe.fr/simon/constants/golomb.txt">The Golomb constant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Golomb-DickmanConstant.html">Golomb-Dickman Constant</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Golomb%E2%80%93Dickman_constant">Golomb-Dickman constant</a>.

%F From _Amiram Eldar_, Aug 25 2020: (Start)

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

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

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

%e 0.62432998854355087...

%p E1:= z-> int(exp(-t)/t, t=z..infinity):

%p lambda:= int(exp(-x-E1(x)), x=0..infinity):

%p s:= convert(evalf(lambda, 130), string):

%p seq(parse(s[n+1]), n=1..120); # _Alois P. Heinz_, Nov 20 2011

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

%o (PARI) intnum(x=0,1-1e-9,exp(-eint1(-log(x)))) \\ _Charles R Greathouse IV_, Jul 28 2015

%o (PARI) default(realprecision, 103);

%o limitnum(n->intnum(x=0, 1-1/n, exp(-eint1(-log(x))))) \\ _Gheorghe Coserea_, Sep 26 2018

%Y Cf. A174974, A174975, A225336.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jun 13 2003