|
%I #24 Oct 16 2019 01:31:20
%S 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,
%T 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,
%U 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
%N 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.
%H Antti Karttunen, <a href="/A123901/b123901.txt">Table of n, a(n) for n = 0..4096</a>
%H 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 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 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>, 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.
%F 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!).
%e a(5) = 4 because (5+3)/gcd(d(5),d(7)) = 8/gcd(2,20) = 8/2 = 4.
%t (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}])
%t (* Second program, faster: *)
%t Table[(n + 3)/Apply[GCD, Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}]], {n, 0, 78}] (* _Michael De Vlieger_, Jul 12 2017 *)
%o (PARI)
%o A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from _Joerg Arndt_, Dec 14 2014
%o A093101(n) = gcd(n!,A000522(n));
%o 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
%Y Cf. A000522, A061354, A093101, A123899, A123900, A124781.
%K easy,nonn
%O 0,1
%A _Jonathan Sondow_, Oct 18 2006
|
