A375781 - OEIS

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

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..11.

Wikipedia, Divergence of the sum of the reciprocals of the primes

EXAMPLE

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

PROG

(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

CROSSREFS

Cf. A374663.

Sequence in context: A259378 A155011 A065406 * A111331 A205668 A118505

Adjacent sequences: A375776 A375778 A375780 * A375782 A375783 A375784

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Aug 28 2024

STATUS

approved