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

1, 2, 6, 15, 105, 1155, 1336335, 892896284280, 398631887241408183843480, 19863422690705846097977473796903171171326157280, 14091270035344566960604487534521565339065390839583445590118556137472614250693240040301050080

Offset: 0

Author

Rémy Sigrist and N. J. A. Sloane, Aug 30 2024

Keywords

• nonn
• frac

Comments

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.

The differences S2(n) - S1(n) are surprisingly small: for n = 1,2,...,34 the values S2(n) - S1(n) are:

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

suggesting the conjecture that they are always 1 except for n = 4 and 6 (compare the Theorem in A374983).

Examples

			The first few fractions are 0/1, 1/2, 5/6, 14/15, 103/105, 1154/1155, 1336333/1336335, 892896284279/892896284280, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..13
• 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. A375781, A375521.

Programs

Maple

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*ithprime(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/(Prime[n]*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n - 1])*Prime[n])];
a[n_] := Denominator[s[n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

Python

from itertools import islice
from math import gcd
from sympy import nextprime
def A375522_gen(): # generator of terms
    p, q, k = 0, 1, 1
    while (k:=nextprime(k)):
        m=q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
        yield q
A375522_list = list(islice(A375522_gen(),11)) # Chai Wah Wu, Aug 30 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 18 2024