A084945 -id:A084945 - cp's OEIS frontend

A174974 Numbers of decimal digits of the Golomb-Dickman constant A084945 that are prime.

6, 27, 57, 60, 1659, 2508

Offset: 1

Eric W. Weisstein, Apr 02 2010

			a(1) = 6 since 624329 is prime
a(2) = 27 since 624329988543550870992936383 is prime

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Golomb-Dickman Constant Digits
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A084945,A174975.

a(5) from David J. Broadhurst, Apr 02 2010

a(6) from Eric W. Weisstein, Apr 03 2010

A174975 Primes in the decimal digits of the Golomb-Dickman constant A084945.

Original entry on oeis.org

624329, 624329988543550870992936383, 624329988543550870992936383100837244179642620180529286973, 624329988543550870992936383100837244179642620180529286973551

Offset: 1

Eric W. Weisstein, Apr 02 2010

Next term has 1659 decimal digits.

David Broadhurst, Titanic Golomb-Dickman prime, digest of 5 messages in primeform Yahoo group, Apr 2 - Apr 9, 2010. [Cached copy]
Eric Weisstein's World of Mathematics, Golomb-Dickman Constant

Cf. A084945, A174974.

A002034 Kempner numbers: smallest positive integer m such that n divides m!.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, 6, 19, 5, 7, 11, 23, 4, 10, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 5, 41, 7, 43, 11, 6, 23, 47, 6, 14, 10, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 8, 13, 11, 67, 17, 23, 7, 71, 6, 73, 37, 10, 19, 11, 13, 79, 6, 9, 41, 83, 7

Offset: 1

N. J. A. Sloane

Sometimes named after Florentin Smarandache, although studied 60 years earlier by Aubrey Kempner and 35 years before that by Lucas.
Kempner originally defined a(1) to be 0, and there are good reasons to prefer that (see Hungerbühler and Specker), but we shall stay with the by-now traditional value a(1) = 1. - N. J. A. Sloane, Jan 02 2021
Kempner gave an algorithm to compute a(n) from the prime factorization of n. Partial solutions were given earlier by Lucas in 1883 and Neuberg in 1887. - Jonathan Sondow, Dec 23 2004
a(n) is the degree of lowest degree monic polynomial over Z that vanishes identically on the integers mod n [Newman].
Smallest k such that n divides product of k consecutive integers starting with n + 1. - Amarnath Murthy, Oct 26 2002
If m and n are any integers with n > 1, then |e - m/n| > 1/(a(n) + 1)! (see Sondow 2006).
Degree of minimal linear recurrence satisfied by Bell numbers (A000110) read modulo n. [Lunnon et al.] - N. J. A. Sloane, Feb 07 2009

			1! = 1, but clearly 8 does not divide 1.
2! = 2, but 8 does not divide 2.
3! = 6, but 8 does not divide 6.
4! = 24, and 8 does divide 24. Hence a(8) = 4.
However, 9 does not divide 24.
5! = 120, but 9 does not divide 120.
6! = 720, and 9 does divide 720. Hence a(9) = 6.

E. Lucas, Question Nr. 288, Mathesis 3 (1883), 232.
R. Muller, Unsolved problems related to Smarandache Function, Number Theory Publishing Company, Phoenix, AZ, ISBN 1-879585-37-5, 1993.
J. Neuberg, Solutions des questions proposées, Question Nr. 288, Mathesis 7 (1887), 68-69.
Donald J. Newman, A Problem Seminar. Problem 17, Springer-Verlag, 1982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Florentin Smarandache, A Function in the Number Theory, Analele Univ. Timisoara, Fascicle 1, Vol. XVIII, 1980, pp. 79-88; Smarandache Function J., Vol. 1, No. 1-3 (1990), pp. 3-17.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 2.5.18 on page 77.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Charles Ashbacher, An Introduction to the Smarandache Function, Erhus Univ. Press, Vail, 62 pages, 1995 [WaybackMachine cached version added by _Felix Fröhlich_, Sep 10 2018].
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 147.
Constantin Dumitrescu, A brief history of the "Smarandache function", Bull. Pure Appl. Sci. Sec. E, Math., Vol. 12, No. 1-2 (1993), pp. 77-82 [WaybackMachine cached version added by _Felix Fröhlich_, Sep 10 2018].
Constantin Dumitrescu and Vasile Seleacu, The Smarandache Function, Erhus Univ. Press, Vail, 137 pages, 1996 [WaybackMachine cached version added by _Felix Fröhlich_, Sep 10 2018].
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), p. 246.
Paul Erdős and Ilias Kastanas, Solution 6674: The smallest factorial that is a multiple of n, Amer. Math. Monthly, Vol. 101, No. 2 (1994) p. 179.
Steven R. Finch, The average value of the Smarandache function, Smarandache Notions Journal, Vol. 10, No. 1-3 (1999), pp. 95-96.
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Norbert Hungerbühler and Ernst Specker, A generalization of the Smarandache Function to several variables, INTEGERS, Vol. 6 (2006), #A23
Aleksandar Ivić, On a problem of Erdős involving the largest prime factor of n, arXiv:math/0311056 [math.NT], 2003-2004.
Aubrey J. Kempner, Miscellanea, The American Mathematical Monthly, Vol. 25, No. 5 (May, 1918), pp. 201-210. (See Section II, Concerning the smallest integer m! divisible by a given integer n.)
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), pp. 45-63.
Xiumei Li and Min Sha, A proof of Sondow's conjecture on the Smarandache function, arXiv preprint arXiv:1907.00370 [math.NT], 2019-2020. Amer. Math. Monthly, 127:10 (2020), 939-943.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., Vol. 35 (1979), pp. 1-16. [From _N. J. A. Sloane_, Feb 07 2009]
Jon Perry, Calculating the Smarandache Numbers, Smarandache Notions Journal, Vol. 14, No. 1 (2004), pp. 124-127.
Project Euler, Problem 549: Divisibility of factorials.
Sebastián Martín Ruiz, A Congruence with Smarandache's Function, Smarandache Notions Journal, Vol. 10, No. 1-3 (1999), pp. 130-132.
József Sándor, The Smarandache minimum and maximum functions, Scientia Magna, Vol. 1, No. 2 (2005), pp. 162-166. See p. 164.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, Vol. 113 (2006), pp. 637-641 and Vol. 114 (2007), p. 659.
Jonathan Sondow and Kyle Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, Vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Smarandache Function.
J. Thompson, A propriety (sic) of smarandache function, Abstract 878-11-758, Notices Amer. Math. Soc., Vol. 14 (1993), p. 41. [Annotated scanned copy]
Wikipedia, Kempner function.
Index entries for sequences related to factorial numbers.

Cf. A000142, A001113, A006530, A007672, A046022, A057109, A064759, A084945, A094371, A094372, A094404, A122378, A122379, A122416, A122417, A248937 (Fermi-Dirac analog: use unique representation of n > 1 as a product of distinct terms of A050376).

See A339594-A339596 for higher-dimensional generalizations.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a002034 1 = 1
a002034 n = fromJust (a092495 n `elemIndex` a000142_list)
-- Reinhard Zumkeller, Aug 24 2011

a:=proc(n) local b: b:=proc(m) if type(m!/n, integer) then m else fi end: [seq(b(m),m=1..100)][1]: end: seq(a(n),n=1..84); # Emeric Deutsch, Aug 01 2005
g:= proc(p,u)
  local i,t;
  t:= 0;
  for i from 1 while t < u do
    t:= t + 1 + padic[ordp](i,p);
  od;
  p*(i-1)
end;
A002034:= x -> max(map(g@op, ifactors(x)[2])); # Robert Israel, Apr 20 2014

Do[m = 1; While[ !IntegerQ[m!/n], m++ ]; Print[m], {n, 85}] (* or for larger n's *)
Kempner[1] := 1; Kempner[n_] := Max[Kempner @@@ FactorInteger[n]]; Kempner[p_, 1] := p; Kempner[p_, alpha_] := Kempner[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; Table[ Kempner[n], {n, 85}] (* Eric W. Weisstein, based on a formula of Kempner's, May 17 2004 *)
With[{facts = Range[100]!}, Flatten[Table[Position[facts, ?(Divisible[#, n] &), {1}, 1], {n, 90}]]] (* _Harvey P. Dale, May 24 2013 *)

a(n)=if(n<0,0,s=1; while(s!%n>0,s++); s)

a(n)=my(s=factor(n)[,1],k=s[#s],f=Mod(k!,n));while(f, f*=k++); k \\ Charles R Greathouse IV, Feb 28 2012

valp(n,p)=my(s);while(n\=p,s+=n);s
K(p,e)=if(e<=p,return(e*p));my(t=e*(p-1)\p*p);while(valp(t+=p,p)Charles R Greathouse IV, Jul 30 2013

from sympy import factorial
def a(n):
    m=1
    while True:
        if factorial(m)%n==0: return m
        else: m+=1
[a(n) for n in range(1, 101)] # Indranil Ghosh, Apr 24 2017

from sympy import factorint
def valp(n, p):
    s = 0
    while n: n //= p; s += n
    return s
def K(p, e):
    if e <= p: return e*p
    t = e*(p-1)//p*p
    while valp(t, p) < e: t += p
    return t
def A002034(n):
    return 1 if n == 1 else max(K(p, e) for p, e in factorint(n).items())
print([A002034(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 09 2022 after Charles R Greathouse IV

A000142(a(n)) = A092495(n). - Reinhard Zumkeller, Aug 24 2011
From Joerg Arndt, Jul 14 2012: (Start)
The following identities were given by Kempner (1918):
a(1) = 1.
a(n!) = n.
a(p) = p for p prime.
a(p1 * p2 * ... * pu) = pu if p1 < p2 < ... < pu are distinct primes.
a(p^k) = p * k for p prime and k <= p.
Let n = p1^e1 * p2^e2 * ... * pu^eu be the canonical factorization of n, then a(n) = max( a(p1^e1), a(p2^e2), ..., a(pu^eu) ).
(End)
Clearly a(n) >= P(n), the largest prime factor of n (= A006530). a(n) = P(n) for almost all n (Erdős and Kastanas 1994, Ivic 2004). The exceptions are A057109. a(n) = P(n) if and only if a(n) is prime because if a(n) > P(n) and a(n) were prime, then since n divides a(n)!, n would also divide (a(n)-1)!, contradicting minimality of a(n). - Jonathan Sondow, Jan 10 2005
If p is prime then a(p^k) = k*p for 0 <= k <= p. Hence it appears also that if n = 2^m * p(1)^e(1) * ... * p(r)^e(r) and if there exists b, 1 <= b <= r, such that Max(2 * m + 2, p(i) * e(i), 1 <= i <= r) = p(b) * e(b) with e(b) <= p(b) then a(n) = e(b) * p(b). E.g.: a(2145986896455317997802121296896) = a(2^10 * 3^3 * 7^9 * 11^9 * 13^8) = 13 * 8 = 104, since 8 * 13 = Max (2 * 10 + 2, 3 * 3, 7 * 9, 11 * 9, 13 * 8) and 8 <= 13. - Benoit Cloitre, Sep 01 2002
It appears that a(2^m - 1) is the largest prime factor of 2^m - 1 (A005420).
a(n!) = n for all n > 0 and a(p) = p if p is prime. - Jonathan Sondow, Dec 23 2004
Conjecture: a(n) = 1 + n - Sum_{k=1..n} Sum_{m=1..n} cos(-2*Pi*k/n*m!)/n. Formula verified for the first 500 terms. - Mats Granvik, Feb 26 2021
Limit_{n->oo} (1/n) * Sum_{k=2..n} log(a(k))/log(k) = A084945 (Finch, 1999). - Amiram Eldar, Jul 04 2021

Error in 45th term corrected by David W. Wilson, May 15 1997

A104350 Partial products of largest prime factors of numbers <= n.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 1260, 2520, 7560, 37800, 415800, 1247400, 16216200, 113513400, 567567000, 1135134000, 19297278000, 57891834000, 1099944846000, 5499724230000, 38498069610000, 423478765710000, 9740011611330000

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Partial Products of A006530: a(n) = Product_{k=1..n} A006530(k).
a(n) = a(n-1)*A006530(n) for n>1, a(1) = 1;
A020639(a(n)) = A040000(n-1), A006530(a(n)) = A007917(n) for n>1.
A001221(a(n)) = A000720(n), A001222(a(n)) = A001477(n-1).
A007947(a(n)) = A034386(n).
a(n) = A000142(n) / A076928(n). [Corrected by Franklin T. Adams-Watters, Oct 30 2006]
In decimal representation: A104351(n) = number of digits of a(n), A104355(n) = number of trailing zeros of a(n).
A104357(n) = a(n) - 1, A104365(n) = a(n) + 1.

Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Publ. Inst. Elie Cartan, Vol. 13, Nancy, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 1..641
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint, arXiv:1202.3670 [math.HO], 2012-2018.
Eric Weisstein's World of Mathematics, Greatest Prime Factor.
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000142, A002110, A006530, A007947, A020639, A046670, A072486, A076928, A104351, A104355, A104357, A104365.

Cf. A001620, A084945.

a104350 n = a104350_list !! (n-1)
a104350_list = scanl1 (*) a006530_list
-- Reinhard Zumkeller, Apr 10 2014

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; Table[A104350[n], {n, 30}] (* G. C. Greubel, May 09 2017 *)
FoldList[Times,Table[FactorInteger[n][[-1,1]],{n,30}]] (* Harvey P. Dale, May 25 2023 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=prod(i=2,n,gpf(i)) \\ Charles R Greathouse IV, Apr 29 2015

first(n)=my(v=vector(n,i,1)); forfactored(k=2,n, v[k[1]]=v[k[1]-1]*vecmax(k[2][,1])); v \\ Charles R Greathouse IV, May 10 2017

log(a(n)) = c * n * log(n) + c * (1-gamma) * n + O(n * exp(-log(n)^(3/8-eps))), where c is the Golomb-Dickman constant (A084945) and gamma is Euler's constant (A001620) (Tenenbaum, 1990). - Amiram Eldar, May 21 2021

More terms from David Wasserman, Apr 24 2008

A271948 Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 20 2016

			0.049469852279228075333485464056253836603725107670028013295315781039...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 289.

Xavier Gourdon, Largest component in random combinatorial structures, Discrete Mathematics 180, 1998, Pages 185-209.

Cf. A084945, A143297, A244067, A244258, A244261, A261873, A271871.

digits = 96; F[x_] := 1 - Exp[-x]/Sqrt[Pi*x] - Erf[Sqrt[x]]; Clear[f, g];
f[m_] := f[m] = 2 NIntegrate[(1 - (1 - F[x])^-1), {x, 0, m}, WorkingPrecision -> digits + 10]; f[m = 100]; f[m = 2 m]; Print["m = ", m]; While[RealDigits[f[m], 10, digits + 5][[1]] != RealDigits[f[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
g[m_] := g[m] = (8/3) NIntegrate[(1 - (1 - F[x])^-1)*x, {x, 0, m}, WorkingPrecision -> digits + 10]; g[m = 100]; g[m = 2 m]; Print["m = ", m]; While[RealDigits[g[m], 10, digits + 5][[1]] != RealDigits[g[m/2], 10, digits + 5][[1]], m = 2 m; Print["m = ", m]];
Join[{0}, RealDigits[g[m] - f[m]^2, 10, digits][[1]]]

A091723 Decimal expansion of the root x of Ei(x)=0, where Ei is the exponential integral.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Feb 01 2004

			0.372507410781366634461991866580119133535689497771654...

Robert Price, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Exponential Integral.
Wikipedia, Exponential integral.

Cf. A070769, A084945.

RealDigits[ x /. FindRoot[ ExpIntegralEi[x] == 0, {x, 1}, WorkingPrecision -> 102]][[1]] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[x /. FindRoot[LogIntegral[Exp[x]]/x, {x, 1/3}, WorkingPrecision -> 105]][[1]] (* Artur Jasinski, Apr 19 2022 *)

solve(x=.3,1,real(eint1(-x))) \\ Charles R Greathouse IV, Apr 14 2014

Equals log(A070769). - Amiram Eldar, Aug 14 2020

Equals root x of li(exp(x)/x)=0 where li(x) is the logarithmic integral. - Artur Jasinski, Apr 19 2022

A244067 Decimal expansion of the Purdom-Williams constant, a constant related to the Golomb-Dickman constant and to the asymptotic evaluation of the expectation of a random function longest cycle length.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2014

			0.78248160099165661501621518806291...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Paul W. Purdom and John H. Williams, Cycle length in a random function, Transactions of the American Mathematical Society, Vol. 133, No. 2 (1968), pp. 547-551.
Eric Weisstein's MathWorld, Golomb-Dickman Constant.
Wikipedia, Golomb-Dickman constant.

Cf. A069998, A084945.

lambda = Integrate[Exp[LogIntegral[x]], {x, 0, 1}]; N[lambda*Sqrt[Pi/2], 99] // RealDigits // First

Equals sqrt(Pi/2)*Integral_{x=0..1} exp(li(x)) dx, where li is the logarithmic integral function.

Equals A069998 * A084945. - Amiram Eldar, Aug 25 2020

A225336 Continued fraction of the Golomb-Dickman constant.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, 3, 1, 1, 28, 5, 2, 3, 13, 1, 4, 3, 1, 1, 1, 1, 1, 2, 1, 43, 1, 1, 1, 2, 3, 1, 38, 1, 2, 5, 48, 2, 66, 1, 1, 1, 1, 2, 1, 9, 6, 13, 1, 5, 2, 38, 18, 1, 9, 2, 2, 12, 491, 1, 1, 19, 1, 1, 1, 1, 2, 1, 19

Offset: 0

Eric W. Weisstein, Jul 25 2013

			lambda = 0.6243299885...
        = [a_0; a_1, a_2, ...]
        = [0; 1, 1, 1, 1, 1, 22, ...]
        = 0 + 1/(1 + 1/(1 + 1/(1 + 1/(22 + 1/(1 + ...))))).

Eric Weisstein's World of Mathematics, Golomb-Dickman Constant Continued Fraction

Cf. A084945 (decimal expansion of the Golomb-Dickman constant).

A244258 Decimal expansion of sqrt(2Pi)log(2), a constant associated with asymptotic evaluation of random mapping statistics.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 24 2014

			1.7374623212723182836603...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4.2 Random Mapping Statistics, p. 288.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002162, A019727, A084945, A244067.

R:= RealField(); Sqrt(2*Pi(R))*Log(2); // G. C. Greubel, Jul 27 2018

RealDigits[Sqrt[2*Pi]*Log[2], 10, 100] // First

sqrt(2*Pi)*log(2) \\ G. C. Greubel, Jul 27 2018

A247398 Decimal expansion of a constant 'v' such that the asymptotic variance of the distribution of the longest cycle given a random n-permutation evaluates as v*n^2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 16 2014

			0.03690783006485220217710700292932764...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.4 Golomb-Dickman Constant, p. 285.

Eric Weisstein's MathWorld, Golomb-Dickman Constant
Wikipedia, Golomb-Dickman constant

Cf. A084945.

evalf(int((x-exp(Ei(-x))*x),x=0..infinity) - int( (1-exp(Ei(-x))),x=0..infinity)^2, 50); # Vaclav Kotesovec, Aug 12 2019

v = NIntegrate[x - E^ExpIntegralEi[-x]*x, {x, 0, Infinity}, WorkingPrecision -> 80] - NIntegrate[1 - E^ExpIntegralEi[-x], {x, 0, Infinity}, WorkingPrecision -> 80]^2; Join[{0}, RealDigits[v, 10, 40] // First]

v = integral_{0..infinity} x-e^Ei(-x)*x dx - (integral_{0..infinity} 1-e^Ei(-x) dx)^2, where Ei is the exponential integral function. [corrected by Vaclav Kotesovec, Aug 12 2019]

More digits from Vaclav Kotesovec, Aug 12 2019

cp's OEIS Frontend

A174974 Numbers of decimal digits of the Golomb-Dickman constant A084945 that are prime.

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A174975 Primes in the decimal digits of the Golomb-Dickman constant A084945.

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A002034 Kempner numbers: smallest positive integer m such that n divides m!.

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A104350 Partial products of largest prime factors of numbers <= n.

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A271948 Decimal expansion of a constant related to the variance of the number of vertices of the largest tree associated with a random mapping on n symbols.

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A091723 Decimal expansion of the root x of Ei(x)=0, where Ei is the exponential integral.

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A244067 Decimal expansion of the Purdom-Williams constant, a constant related to the Golomb-Dickman constant and to the asymptotic evaluation of the expectation of a random function longest cycle length.

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A244258 Decimal expansion of sqrt(2Pi)log(2), a constant associated with asymptotic evaluation of random mapping statistics.