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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.

%I A375531 #30 Apr 22 2025 09:08:55
%S A375531 2,5,61,14641,1071721201,6891517989606967201,
%T A375531 332451141407535184183280941400379650401,
%U A375531 884190091385383640998709844252171404846723555306050253676905585566612798483201
%N 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.
%H A375531 Andrew Howroyd, <a href="/A375531/b375531.txt">Table of n, a(n) for n = 1..11</a>
%H A375531 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 A375531 a(n) = n!*A375532(n-1) + 1.
%p A375531 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/a(n)) end:
%p A375531 a:= proc(n) a(n):= 1+floor(n!/(1-s(n-1))) end:
%p A375531 seq(a(n), n=1..8);  # _Alois P. Heinz_, Oct 18 2024
%t A375531 s[n_] := s[n] = If[n == 0, 0, s[n - 1] + n!/a[n]];
%t A375531 a[n_] := a[n] = 1 + Floor[n!/(1 - s[n - 1])];
%t A375531 Table[a[n], {n, 1, 8}] (* _Jean-François Alcover_, Apr 22 2025, after _Alois P. Heinz_ *)
%o A375531 (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 A375531 a(n)={B(vector(n, k, k!))[n]} \\ _Andrew Howroyd_, Sep 04 2024
%Y A375531 Cf. A000142, A374663, A374983/A375516, A375532.
%K A375531 nonn
%O A375531 1,1
%A A375531 _N. J. A. Sloane_, Sep 04 2024