A135506 a(n) = x(n+1)/x(n) - 1 where x(1)=1 and x(k) = x(k-1) + lcm(x(k-1),k). Here x(n) = A135504(n).

2, 1, 2, 5, 1, 7, 1, 1, 5, 11, 1, 13, 1, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 5, 13, 1, 1, 29, 1, 31, 1, 11, 17, 1, 1, 37, 1, 13, 1, 41, 1, 43, 1, 1, 23, 47, 1, 1, 1, 17, 13, 53, 1, 1, 1, 1, 29, 59, 1, 61, 1, 1, 1, 13, 1, 67, 1, 23, 1, 71, 1, 73, 1, 1, 1, 1, 13, 79, 1, 1, 41, 83, 1, 1, 43, 29, 1, 89

Offset: 1

Author

Benoit Cloitre, Feb 09 2008

Keywords

• nonn

Comments

This sequence has properties related to primes. For instance: terms consist of 1's or primes only; if 3 never occurs, any prime p occurs finitely many times.

All prime numbers 'p' from the sequence A014963(n), which equals A003418(n+1)/A003418(n), are in a(n-1) = p. - Eric Desbiaux, Jan 11 2015

For any prime p > 3, a(p-1) = p. Also a(n) is not 3 for any n. All terms but a(1) and a(3) are odd, and probably all of them are not composite numbers; this is strongly related to a strong version of Linnik's Theorem (see Ruiz-Cabello link). - Serafín Ruiz-Cabello, Sep 30 2015

Per the prior comment, the distinct prime terms correspond to A045344. This is every prime except for 3. - Bill McEachen, Sep 12 2022

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Robert Israel)
• Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, YouTube video, 2023. (See especially the last section beginning at 20:08).
• Serafín Ruiz-Cabello, On the use of the lowest common multiple to build a prime-generating recurrence, arXiv:1504.05041 [math.CO], 2015.

Crossrefs

Cf. A045344, A135504, A361460, A361461 (positions of 1's), A361462 [= a(n) mod 4], A361463, A361464, A361470.

Cf. also A106108.

Programs

Maple

x[1]:= 1;
for n from 2 to 101 do
  x[n]:= x[n-1] + ilcm(x[n-1],n);
  a[n-1]:= x[n]/x[n-1]-1;
od:
seq(a[n],n=1..100); # Robert Israel, Jan 11 2015

Mathematica

a[n_] := x[n+1]/x[n] - 1; x[1] = 1; x[k_] := x[k] = x[k-1] + LCM[x[k-1], k]; Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Jan 08 2013 *)

PARI

x1=1;for(n=2,40,x2=x1+lcm(x1,n);t=x1;x1=x2;print1(x2/t-1,","))

Python

from itertools import count, islice
from math import lcm
def A135506_gen(): # generator of terms
    x = 1
    for n in count(2):
        y = x+lcm(x,n)
        yield y//x-1
        x = y
A135506_list = list(islice(A135506_gen(),20)) # Chai Wah Wu, Mar 13 2023

Formula

a(n) = A135504(n+1)/A135504(n) - 1.

a(n) = (n+1) / A361470(n). - Antti Karttunen, Mar 26 2023

Extensions

References to A135504 added by Antti Karttunen, Mar 07 2023