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A374983 a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A374663(k)).

%I A374983 #51 Apr 22 2025 08:59:37
%S A374983 0,1,3,11,47,1199,241199,9696481199,11752718467440661199,
%T A374983 15347376819435640471203267700016821199,
%U A374983 23554197523775043569951631809272942045755944094320810352530343995293765199
%N A374983 a(n) is the numerator of Sum_{k = 1..n} 1 / (k*A374663(k)).
%C A374983 For the denominators see A375516 and A375517.
%C A374983 For n = 1..36, Sum_{k = 1..n} 1 / (k*A374663(k)) = a(n) / (1 + a(n)). In fact this holds for all n >= 1.
%C A374983 Theorem: Let S_n = Sum_{k = 1..n} 1 / (k*A374663(k)) and let r_n = 1 - S_n. Then for  n > 1, r_n is the inverse of a positive integer, say d_n; d_{n+1} is divisible by d_n; and d_n is divisible by all positive integers < n. (See Sigrist link for proof; d_n is given in A375516.)
%H A374983 N. J. A. Sloane, <a href="/A374983/b374983.txt">Table of n, a(n) for n = 0..14</a>
%H A374983 Rémy Sigrist, <a href="/A374983/a374983.txt">Proof of Theorem</a>, Aug 26 2024, revised Sep 01 2024.
%H A374983 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]
%e A374983 For n = 3: A374663(1) = A374663(2) = A374663(3) = 2, 1/(1*2) + 1/(2*2) + 1/(3*2) = 11/12, so a(3) = 11.
%p A374983 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
%p A374983 b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
%p A374983 a:= n-> numer(s(n)):
%p A374983 seq(a(n), n=0..10);  # _Alois P. Heinz_, Oct 18 2024
%t A374983 s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(n*b[n])];
%t A374983 b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*n)];
%t A374983 a[n_] := Numerator[s[n]];
%t A374983 Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Apr 22 2025, after _Alois P. Heinz_ *)
%o A374983 (PARI) { print1 (0); t = 0; for (n = 1, 10, for (v = c=ceil(1/(n*(1-t))), oo, if (t + 1/(n*v) < 1, t += 1/(n*v); print1 (", " numerator(t)); break;););); }
%o A374983 (Python)
%o A374983 from itertools import count, islice
%o A374983 from math import gcd
%o A374983 def A374983_gen(): # generator of terms
%o A374983     p, q = 0, 1
%o A374983     for k in count(1):
%o A374983         yield p
%o A374983         m = q//(k*(q-p))+1
%o A374983         p, q = p*k*m+q, k*m*q
%o A374983         p //= (r:=gcd(p,q))
%o A374983         q //= r
%o A374983 A374983_list = list(islice(A374983_gen(),11)) # _Chai Wah Wu_, Aug 28 2024
%Y A374983 Cf. A374663, A375516 (denominators), A375517.
%K A374983 nonn,frac
%O A374983 0,3
%A A374983 _Rémy Sigrist_, Aug 04 2024