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 A102468 #30 Feb 16 2025 08:32:55
%S A102468 1,2,5,4,13,163,103,137,863,98641,10687,31469,1540901,522787,5441,
%T A102468 226871807,13619,1276861,414026539,2124467,12670743557,838025081381,
%U A102468 44659157,323895443,337310723185584470837549,54352957
%N A102468 a(n)! is the smallest factorial divisible by the numerator of Sum_{k=0...n} 1/k!, with a(0) = 1.
%C A102468 It appears that a(n) = A102469(n) (largest prime factor of the same numerator) except when n = 3. The smallest factorial divisible by the corresponding denominator is n!. Omitting the 0th term in the sum, it appears that the Kempner number (A002034) and the largest prime factor, of the numerator of Sum_{k=1...n} 1/k! are both equal to A096058(n).
%C A102468 The Mathematica program given below was used to generate the sequence. If the numerator of Sum_{k=0...n}(1/k!) is squarefree, the program prints the value of the numerator's largest prime factor, which must equal a(n). Otherwise, the program prints the complete factorization of the numerator so a(n) can be determined by inspection. - _Ryan Propper_, Jul 31 2005
%H A102468 A. J. Kempner, <a href="http://www.jstor.org/stable/2972639">Miscellanea</a>, Amer. Math. Monthly, 25 (1918), 201-210 [ See Section II, "Concerning the smallest integer m! divisible by a given integer n". ]
%H A102468 J. Sondow, <a href="http://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 A102468 J. Sondow, <a href="http://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 A102468 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 A102468 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">Smarandache Function.</a>
%H A102468 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers.</a>
%F A102468 a(n) = A002034(A061354(n)).
%e A102468 Sum_{k=0...3} 1/k! = 8/3 and 4! is the smallest factorial divisible by 8, so a(3) = 4.
%t A102468 Do[l = FactorInteger[Numerator[Sum[1/k!, {k, 0, n}]]]; If[Length[l] == Plus @@ Last /@ l, Print[Max[First /@ l]], Print[l]], {n, 1, 30}] (* _Ryan Propper_, Jul 31 2005 *)
%t A102468 nmax = 30; Clear[a]; Do[f = FactorInteger[ Numerator[ Sum[1/k!, {k, 0, n}] ] ]; a[n] = If[Length[f] == Total[f[[All, 2]] ], Max[f[[All, 1]] ], f[[-1, 1]] ], {n, 0, nmax}]; a[3] = 4; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Sep 16 2015, adapted from _Ryan Propper_'s script *)
%o A102468 (PARI) a(n) = {j = 1; s = numerator(sum(k=0, n, 1/k!)); while (j! % s, j++); j;} \\ _Michel Marcus_, Sep 16 2015
%Y A102468 Cf. A102469, A000522, A002034, A061354, A096058, A007917.
%K A102468 nonn
%O A102468 0,2
%A A102468 _Jonathan Sondow_, Jan 09 2005
%E A102468 More terms from _Ryan Propper_, Jul 31 2005