This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102469 #27 Feb 16 2025 08:32:55
%S A102469 1,2,5,2,13,163,103,137,863,98641,10687,31469,1540901,522787,5441,
%T A102469 226871807,13619,1276861,414026539,2124467,12670743557,838025081381,
%U A102469 44659157,323895443,337310723185584470837549,54352957,11301647941785046703319941,102505951982728548829
%N A102469 Largest prime factor of numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.
%C A102469 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).
%H A102469 Daniel Suteu, <a href="/A102469/b102469.txt">Table of n, a(n) for n = 0..74</a>
%H A102469 J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.
%H A102469 J. Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.
%H A102469 J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
%H A102469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>
%H A102469 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%F A102469 a(n) = A006530(A061354(n)).
%e A102469 Sum_{k=0...3} 1/k! = 8/3 and 2 is the largest prime factor 8, so a(3) = 2.
%t A102469 FactorInteger[#][[-1,1]]&/@Numerator[Accumulate[1/Range[0,30]!]] (* _Harvey P. Dale_, Nov 14 2012 *)
%o A102469 (PARI) a(n) = if(n==0, return(1)); vecmax(factor(numerator(sum(k=0, n, 1/k!)))[,1]); \\ _Daniel Suteu_, Jun 09 2022
%Y A102469 Cf. A102468, A096058, A006530, A061354, A000522, A007917.
%K A102469 nonn
%O A102469 0,2
%A A102469 _Jonathan Sondow_, Jan 09 2005