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A124780 a(n) = gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

%I A124780 #21 Jun 14 2022 13:49:39
%S A124780 1,2,5,2,1,2,1,10,1,2,13,2,5,2,1,2,1,10,1,2,1,2,5,26,1,2,1,10,1,2,1,2,
%T A124780 5,2,37,2,13,10,1,2,1,2,5,2,1,2,1,10,1,26,1,2,5,2,1,2,1,10,1,2,1,2,65,
%U A124780 2,1,2,1,10,1,2,1,74,5,2,1,26,1,10,1,2,1,2,5,2,1,2,1,10,13,2,1,2,5,2,1,2,1
%N A124780 a(n) = gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.
%C A124780 a(n) divides n+3 because A(n+2) = (n+2)(n+1)*A(n) + n+3.
%H A124780 Antti Karttunen, <a href="/A124780/b124780.txt">Table of n, a(n) for n = 0..4096</a>
%H A124780 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 A124780 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 A124780 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 A124780 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A124780 a(n) = gcd(A000522(n), A000522(n+2)) = (n+3)/A124782(n)
%e A124780 a(2) = gcd(A(2), A(4)) = gcd(5, 65) = 5.
%t A124780 (A[n_] := Sum[n!/k!, {k, 0, n}]; Table[GCD[A[n],A[n+2]], {n,0,100}])
%t A124780 GCD[#[[1]],#[[3]]]&/@Partition[Table[Sum[n!/k!,{k,0,n}],{n,0,100}],3,1] (* _Harvey P. Dale_, Jun 14 2022 *)
%o A124780 (PARI)
%o A124780 A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from _Joerg Arndt_, Dec 14 2014
%o A124780 A124780(n) = gcd(A000522(n),A000522(n+2)); \\ _Antti Karttunen_, Jul 07 2017
%Y A124780 Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124781, A124782.
%K A124780 nonn
%O A124780 0,2
%A A124780 _Jonathan Sondow_, Nov 07 2006