

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
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OFFSET

0,1


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) 637641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 20072010.
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) = (n+3)/A124781(n) = (n+3)/gcd(A093101(n),A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n1)+...+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[n1]+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

Cf. A000522, A061354, A093101, A123899, A123900, A124781.
Sequence in context: A276737 A270027 A271726 * A214682 A093395 A176774
Adjacent sequences: A123898 A123899 A123900 * A123902 A123903 A123904


KEYWORD

easy,nonn


AUTHOR

Jonathan Sondow, Oct 18 2006


STATUS

approved



