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Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} k!/a(k) < 1.
19

%I #26 Oct 20 2024 23:48:41

%S 2,5,61,14641,1071721201,6891517989606967201,

%T 332451141407535184183280941400379650401,

%U 884190091385383640998709844252171404846723555306050253676905585566612798483201

%N Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} k!/a(k) < 1.

%H Andrew Howroyd, <a href="/A375531/b375531.txt">Table of n, a(n) for n = 1..11</a>

%H N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=3RAYoaKMckM">A Nasty Surprise in a Sequence and Other OEIS Stories</a>, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane85BD.pdf">Slides</a> [Mentions this sequence]

%F a(n) = n!*A375532(n-1) + 1.

%p s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/a(n)) end:

%p a:= proc(n) a(n):= 1+floor(n!/(1-s(n-1))) end:

%p seq(a(n), n=1..8); # _Alois P. Heinz_, Oct 18 2024

%o (PARI) B(u)={my(v=vector(#u)); my(r=1); for(i=1, #u, my(t=floor(u[i]/r)+1); v[i]=t; r-=u[i]/t); v}

%o a(n)={B(vector(n, k, k!))[n]} \\ _Andrew Howroyd_, Sep 04 2024

%Y Cf. A000142, A374663, A374983/A375516, A375532.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Sep 04 2024