A375781 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (prime(k)*a(k)) < 1 (where prime(k) denotes the k-th prime number).
1, 1, 2, 3, 5, 89, 39304, 46994541278, 17331821184409051471456, 684945610024339520619912889548385212804350252, 454557097914340869696918952726502107711786801276885341616727617337826266151394840009711293
Offset: 1
Keywords
Examples
The first terms, alongside the corresponding sums, are:
n a(n) Sum_{k=1..n} 1/(prime(k)*a(k))
- ----- ------------------------------
1 1 1/2
2 1 5/6
3 2 14/15
4 3 103/105
5 5 1154/1155
6 89 1336333/1336335
7 39304 892896284279/892896284280
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..14
- 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]
- Wikipedia, Divergence of the sum of the reciprocals of the primes
Programs
-
Maple
s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*a(n))) end: a:= proc(n) a(n):= 1+floor(1/((1-s(n-1))*ithprime(n))) end: seq(a(n), n=1..11); # Alois P. Heinz, Oct 18 2024
-
Mathematica
s[n_] := s[n] = If[n == 0, 0, s[n-1] + 1/(Prime[n]*a[n])]; a[n_] := a[n] = 1 + Floor[1/((1 - s[n-1])*Prime[n])]; Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Mar 19 2025, after Alois P. Heinz *) -
PARI
{ r = 1; forprime (p = 2, prime(11), print1 (a = floor(1/(r*p)) + 1", "); r -= 1 / (a*p);); } -
Python
from itertools import islice from math import gcd from sympy import nextprime def A375781_gen(): # generator of terms p, q, k = 0, 1, 1 while (k:=nextprime(k)): yield (m:=q//(k*(q-p))+1) p, q = p*k*m+q, k*m*q p //= (r:=gcd(p,q)) q //= r A375781_list = list(islice(A375781_gen(),11)) # Chai Wah Wu, Aug 30 2024
Comments