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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).

%I A375781 #19 Mar 19 2025 10:13:11
%S A375781 1,1,2,3,5,89,39304,46994541278,17331821184409051471456,
%T A375781 684945610024339520619912889548385212804350252,
%U A375781 454557097914340869696918952726502107711786801276885341616727617337826266151394840009711293
%N 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).
%C A375781 The sum of the reciprocals of the primes diverges. We divide each of its terms in such a way as to have a series bounded by 1.
%H A375781 Alois P. Heinz, <a href="/A375781/b375781.txt">Table of n, a(n) for n = 1..14</a>
%H A375781 N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=3RAYoaKMckM">A Nasty Surprise in a Sequence and Other OEIS Stories</a>, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane85BD.pdf">Slides</a> [Mentions this sequence]
%H A375781 Wikipedia, <a href="https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes">Divergence of the sum of the reciprocals of the primes</a>
%e A375781 The first terms, alongside the corresponding sums, are:
%e A375781   n  a(n)   Sum_{k=1..n} 1/(prime(k)*a(k))
%e A375781   -  -----  ------------------------------
%e A375781   1      1  1/2
%e A375781   2      1  5/6
%e A375781   3      2  14/15
%e A375781   4      3  103/105
%e A375781   5      5  1154/1155
%e A375781   6     89  1336333/1336335
%e A375781   7  39304  892896284279/892896284280
%p A375781 s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(ithprime(n)*a(n))) end:
%p A375781 a:= proc(n) a(n):= 1+floor(1/((1-s(n-1))*ithprime(n))) end:
%p A375781 seq(a(n), n=1..11);  # _Alois P. Heinz_, Oct 18 2024
%t A375781 s[n_] := s[n] = If[n == 0, 0, s[n-1] + 1/(Prime[n]*a[n])];
%t A375781 a[n_] := a[n] = 1 + Floor[1/((1 - s[n-1])*Prime[n])];
%t A375781 Table[a[n], {n, 1, 11}] (* _Jean-François Alcover_, Mar 19 2025, after _Alois P. Heinz_ *)
%o A375781 (PARI) { r = 1; forprime (p = 2, prime(11), print1 (a = floor(1/(r*p)) + 1", "); r -= 1 / (a*p);); }
%o A375781 (Python)
%o A375781 from itertools import islice
%o A375781 from math import gcd
%o A375781 from sympy import nextprime
%o A375781 def A375781_gen(): # generator of terms
%o A375781     p, q, k = 0, 1, 1
%o A375781     while (k:=nextprime(k)):
%o A375781         yield (m:=q//(k*(q-p))+1)
%o A375781         p, q = p*k*m+q, k*m*q
%o A375781         p //= (r:=gcd(p,q))
%o A375781         q //= r
%o A375781 A375781_list = list(islice(A375781_gen(),11)) # _Chai Wah Wu_, Aug 30 2024
%Y A375781 Cf. A000040, A374663.
%K A375781 nonn
%O A375781 1,3
%A A375781 _Rémy Sigrist_, Aug 28 2024