OFFSET
0,2
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).
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]
EXAMPLE
The first few fractions are 0/1, 1/2, 5/6, 14/15, 103/105, 1154/1155, 1336333/1336335, 892896284279/892896284280, ...
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
PROG
(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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Rémy Sigrist and N. J. A. Sloane, Aug 30 2024
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Oct 18 2024
STATUS
approved