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 A124781 a(n) = gcd(A093101(n), A093101(n+2)) where A093101(n) = gcd(n!, A(n)) and A(n) = A000522(n) = Sum_{k=0..n} n!/k!). 5
 1, 1, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 26, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) divides n+3 because A(n+2) = (n+2)(n+1)*A(n) + n+3. LINKS Antti Karttunen, Table of n, a(n) for n = 0..4096 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. FORMULA a(n) = gcd(A093101(n), A093101(n+2)) = (n+3)/A123901(n). a(n) = gcd(A(n), A(n+2), n!) where A(n)=1+n+n(n-1)+...+n!. - Jonathan Sondow, Nov 13 2006 EXAMPLE a(3) = gcd(d(3),d(5)) = gcd(gcd(3!,16), gcd(5!,326)) = gcd(2,2) = 2. MATHEMATICA (A[n_] := Sum[n!/k!, {k, 0, n}]; d[n_] := GCD[n!, A[n]]; Table[GCD[d[n], d[n+2]], {n, 0, 100}]) (* Second program, faster: *) Table[GCD @@ Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}], {n, 0, 96}] (* Michael De Vlieger, Jul 12 2017 *) PROG (PARI) A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014 A093101(n) = gcd(n!, A000522(n)); m1=m2=1; for(n=0, 4096, m=m1; m1=m2; m2 = A093101(n+2); m124781 = gcd(m, m2); write("b093101.txt", n, " ", m); write("b124781.txt", n, " ", m124781); write("b123901.txt", n, " ", (n+3)/m124781)); \\ Antti Karttunen, Jul 12 2017 CROSSREFS Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124782. Sequence in context: A066772 A104060 A062347 * A124151 A110179 A071559 Adjacent sequences:  A124778 A124779 A124780 * A124782 A124783 A124784 KEYWORD nonn AUTHOR Jonathan Sondow, Nov 07 2006 EXTENSIONS Replaced d(n) in the name with A093101(n). - Antti Karttunen, Jul 12 2017 STATUS approved

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Last modified May 23 07:30 EDT 2019. Contains 323508 sequences. (Running on oeis4.)