A102469 Largest prime factor of numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.

1, 2, 5, 2, 13, 163, 103, 137, 863, 98641, 10687, 31469, 1540901, 522787, 5441, 226871807, 13619, 1276861, 414026539, 2124467, 12670743557, 838025081381, 44659157, 323895443, 337310723185584470837549, 54352957, 11301647941785046703319941, 102505951982728548829

Offset: 0

Author

Jonathan Sondow, Jan 09 2005

Keywords

• nonn

Comments

It appears that a(n) = A102468(n) (Smarandache number of the same numerator) except when n = 3. The largest prime factor of the corresponding denominator is A007917(n) for n > 1. Omitting the 0th term in the sum, it appears that the largest prime factor and the Kempner number A002034, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).

Examples

			Sum_{k=0...3} 1/k! = 8/3 and 2 is the largest prime factor 8, so a(3) = 2.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 0..74
• J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
• J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
• J. Sondow and K. 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.
• Eric Weisstein's World of Mathematics, Greatest Prime Factor
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A102468, A096058, A006530, A061354, A000522, A007917.

Programs

Mathematica

FactorInteger[#][[-1,1]]&/@Numerator[Accumulate[1/Range[0,30]!]] (* Harvey P. Dale, Nov 14 2012 *)

PARI

a(n) = if(n==0, return(1)); vecmax(factor(numerator(sum(k=0, n, 1/k!)))[,1]); \\ Daniel Suteu, Jun 09 2022

Formula

a(n) = A006530(A061354(n)).