A375531 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.

2, 5, 61, 14641, 1071721201, 6891517989606967201, 332451141407535184183280941400379650401, 884190091385383640998709844252171404846723555306050253676905585566612798483201

Offset: 1

Author

N. J. A. Sloane, Sep 04 2024

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..11
• N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Crossrefs

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

Programs

Maple

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/a(n)) end:
a:= proc(n) a(n):= 1+floor(n!/(1-s(n-1))) end:
seq(a(n), n=1..8);  # Alois P. Heinz, Oct 18 2024

Mathematica

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + n!/a[n]];
a[n_] := a[n] = 1 + Floor[n!/(1 - s[n - 1])];
Table[a[n], {n, 1, 8}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

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}
a(n)={B(vector(n, k, k!))[n]} \\ Andrew Howroyd, Sep 04 2024

Formula

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