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A375532 a(n) is the denominator of Sum_{k = 1..n} k! / A375531(k).

%I A375532 #34 Dec 10 2024 08:58:04
%S A375532 1,2,10,610,8931010,9571552763343010,
%T A375532 65962528057050631782397012182615010,
%U A375532 21929317742693046651753716375301870159888977066122278116986745673775119010
%N A375532 a(n) is the denominator of Sum_{k = 1..n} k! / A375531(k).
%H A375532 Alois P. Heinz, <a href="/A375532/b375532.txt">Table of n, a(n) for n = 0..10</a>
%H A375532 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 A375532 a(n+1) = (n+1)!*a(n)^2 + a(n), with a(1) = 2.
%e A375532 The first few sums are 0/1, 1/2, 9/10, 609/610, 8931009/8931010, 9571552763343009/9571552763343010, ...
%p A375532 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+n!/b(n)) end:
%p A375532 b:= proc(n) b(n):= 1+floor(n!/(1-s(n-1))) end:
%p A375532 a:= n-> denom(s(n)):
%p A375532 seq(a(n), n=0..7);  # _Alois P. Heinz_, Oct 18 2024
%t A375532 s[n_] := s[n] = If[n == 0, 0, s[n-1] + n!/b[n]];
%t A375532 b[n_] := b[n] = 1 + Floor[n!/(1 - s[n-1])];
%t A375532 a[n_] := Denominator[s[n]];
%t A375532 Table[a[n], {n, 0, 7}] (* _Jean-François Alcover_, Dec 10 2024, after _Alois P. Heinz_ *)
%Y A375532 Cf. A000142, A374663, A374983/A375516, A375531.
%K A375532 nonn,frac
%O A375532 0,2
%A A375532 _N. J. A. Sloane_, Sep 04 2024
%E A375532 a(0)=1 prepended by _Alois P. Heinz_, Oct 18 2024