A375521 -id:A375521 - cp's OEIS frontend

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

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

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).

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

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]

Cf. A375781, A375521.

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

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 *)

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

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

A375528 a(n) = denominator of Sum_{k = 1..n} 1 / (A000959(k)*A375527(k)).

Original entry on oeis.org

1, 2, 6, 42, 630, 57330, 219172590, 2287458514758690, 523246645674205487113407810300, 34223381526163442974989472671319545640510650941743506071550, 65068880171408068403202506207461768112305307530373013598603234255112994800902512713302330140957468591804616490482800

Offset: 1

N. J. A. Sloane, Sep 01 2024

nonn

The first few sums S(n) = Sum_{k = 1..n} 1/(A000959(k)*A375527(k)) are: 1/2, 5/6, 41/42, 629/630, 57329/57330,
 219172589/219172590, 2287458514758689/2287458514758690,
 523246645674205487113407810299/523246645674205487113407810300, ..., and the first 10 or 11 of these sums have the form (c-1)/c, where c is an integer. The present sequence gives the denominators.
For the harmonic series analog, A374663, Rémy Sigrist has shown that all the partial sums have that form (see A374983), and for the prime number analog, A375581, it seems that all partial sums except for n = 4 and 6 have this property (see A375521/A375522).

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]

Cf. A000959, A375527, A374683, A374983/A375516, A375582, A375521/A375522.

cp's OEIS Frontend

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

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