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A375522 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A375781(k)).

%I A375522 #25 Apr 22 2025 09:08:48
%S A375522 1,2,6,15,105,1155,1336335,892896284280,398631887241408183843480,
%T A375522 19863422690705846097977473796903171171326157280,
%U A375522 14091270035344566960604487534521565339065390839583445590118556137472614250693240040301050080
%N A375522 a(n) is the denominator of Sum_{k = 1..n} 1 / (k*A375781(k)).
%C A375522 Let S(n) = Sum_{k = 1..n} 1 / (k*A375781(k)) = S1(n)/S2(n) when reduced to lowest terms, where S1(n) = A375521(n), S2(n) = the present sequence.
%C A375522 The differences S2(n) - S1(n) are surprisingly small: for n = 1,2,...,34 the values S2(n) - S1(n) are:
%C A375522 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%C A375522 suggesting the conjecture that they are always 1 except for n = 4 and 6 (compare the Theorem in A374983).
%H A375522 Alois P. Heinz, <a href="/A375522/b375522.txt">Table of n, a(n) for n = 0..13</a>
%H A375522 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 A375522 The first few fractions are 0/1, 1/2, 5/6, 14/15, 103/105, 1154/1155, 1336333/1336335, 892896284279/892896284280, ...
%p A375522 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*b(n))) end:
%p A375522 b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*ithprime(n))) end:
%p A375522 a:= n-> denom(s(n)):
%p A375522 seq(a(n), n=0..10);  # _Alois P. Heinz_, Oct 18 2024
%t A375522 s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(Prime[n]*b[n])];
%t A375522 b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*Prime[n])];
%t A375522 a[n_] := Denominator[s[n]];
%t A375522 Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Apr 22 2025, after _Alois P. Heinz_ *)
%o A375522 (Python)
%o A375522 from itertools import islice
%o A375522 from math import gcd
%o A375522 from sympy import nextprime
%o A375522 def A375522_gen(): # generator of terms
%o A375522     p, q, k = 0, 1, 1
%o A375522     while (k:=nextprime(k)):
%o A375522         m=q//(k*(q-p))+1
%o A375522         p, q = p*k*m+q, k*m*q
%o A375522         p //= (r:=gcd(p,q))
%o A375522         q //= r
%o A375522         yield q
%o A375522 A375522_list = list(islice(A375522_gen(),11)) # _Chai Wah Wu_, Aug 30 2024
%Y A375522 Cf. A375781, A375521.
%K A375522 nonn,frac
%O A375522 0,2
%A A375522 _Rémy Sigrist_ and _N. J. A. Sloane_, Aug 30 2024
%E A375522 a(0)=1 prepended by _Alois P. Heinz_, Oct 18 2024