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

1, 2, 4, 12, 48, 1200, 241200, 9696481200, 11752718467440661200, 15347376819435640471203267700016821200, 23554197523775043569951631809272942045755944094320810352530343995293765200

Offset: 0

Author

N. J. A. Sloane, Aug 19 2024

Keywords

• nonn
• frac

Comments

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.

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.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..14
• Rémy Sigrist, Proof of theorem about A374983 and the present sequence, Aug 26 2024, revised Sep 01 2024.
• N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Crossrefs

Cf. A374663, A374983.

See A375517 for a(n)/n and A375791 for a(n+1)/a(n).

Programs

Maple

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
a:= n-> denom(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 18 2024

Mathematica

s[n_] := s[n] = If[n == 0, 0, s[n-1] + 1/(n*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n-1])*n)];
a[n_] := Denominator[s[n]];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 10 2024, after Alois P. Heinz *)

Python

from itertools import count, islice
from math import gcd
def A375516_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        yield q
        m = q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A375516_list = list(islice(A375516_gen(),11)) # Chai Wah Wu, Aug 28 2024