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A123901
a(n) = (n+3)/gcd(d(n), d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0..n} 1/k! to lowest terms.
8
3, 4, 5, 3, 7, 4, 9, 1, 11, 6, 13, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 1, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 37, 19, 3, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 2, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 1, 33, 67, 34, 69, 7, 71, 36, 73, 1, 15, 38, 77, 3, 79, 8, 81
OFFSET
0,1
LINKS
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 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, arXiv:0709.0671 [math.NT], 2007-2009; 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) = (n+3)/A124781(n) = (n+3)/gcd(A093101(n),A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n-1)+...+n!).
EXAMPLE
a(5) = 4 because (5+3)/gcd(d(5),d(7)) = 8/gcd(2,20) = 8/2 = 4.
MATHEMATICA
(A[n_] := If[n==0, 1, n*A[n-1]+1]; d[n_] := GCD[A[n], n! ]; Table[(n+3)/GCD[d[n], d[n+2]], {n, 0, 79}])
(* Second program, faster: *)
Table[(n + 3)/Apply[GCD, Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}]], {n, 0, 78}] (* 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
KEYWORD
easy,nonn
AUTHOR
Jonathan Sondow, Oct 18 2006
STATUS
approved