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A375516 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A374663(k)).

%I A375516 #42 Dec 10 2024 10:00:59
%S A375516 1,2,4,12,48,1200,241200,9696481200,11752718467440661200,
%T A375516 15347376819435640471203267700016821200,
%U A375516 23554197523775043569951631809272942045755944094320810352530343995293765200
%N A375516 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A374663(k)).
%C A375516 In fact a(n) = A374983(n) + 1 (see the proof in A374983), but this was unproved when this sequence was created, and in any case the prime factors of A374983(n) and a(n) are both of interest, so both sequences are included in the OEIS. Both sequences grow doubly exponentially. See also A375791.
%C A375516 One might be led to conjecture that the last 4 digits of the numbers from a(5) onwards are always 1200, but _Rémy Sigrist_ has observed that this does not hold for a(10) = 23554197523775043569951631809272942045755944094320810352530343995293765200.
%H A375516 N. J. A. Sloane, <a href="/A375516/b375516.txt">Table of n, a(n) for n = 0..14</a>
%H A375516 Rémy Sigrist, <a href="/A374983/a374983.txt">Proof of theorem about A374983 and the present sequence</a>, Aug 26 2024, revised Sep 01 2024.
%H A375516 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]
%p A375516 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
%p A375516 b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
%p A375516 a:= n-> denom(s(n)):
%p A375516 seq(a(n), n=0..10);  # _Alois P. Heinz_, Oct 18 2024
%t A375516 s[n_] := s[n] = If[n == 0, 0, s[n-1] + 1/(n*b[n])];
%t A375516 b[n_] := b[n] = 1 + Floor[1/((1 - s[n-1])*n)];
%t A375516 a[n_] := Denominator[s[n]];
%t A375516 Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Dec 10 2024, after _Alois P. Heinz_ *)
%o A375516 (Python)
%o A375516 from itertools import count, islice
%o A375516 from math import gcd
%o A375516 def A375516_gen(): # generator of terms
%o A375516     p, q = 0, 1
%o A375516     for k in count(1):
%o A375516         yield q
%o A375516         m = q//(k*(q-p))+1
%o A375516         p, q = p*k*m+q, k*m*q
%o A375516         p //= (r:=gcd(p,q))
%o A375516         q //= r
%o A375516 A375516_list = list(islice(A375516_gen(),11)) # _Chai Wah Wu_, Aug 28 2024
%Y A375516 Cf. A374663, A374983.
%Y A375516 See A375517 for a(n)/n and A375791 for a(n+1)/a(n).
%K A375516 nonn,frac
%O A375516 0,2
%A A375516 _N. J. A. Sloane_, Aug 19 2024