OFFSET
1,1
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.
FORMULA
a(n) = (n+1) / A361470(n). - Antti Karttunen, Mar 26 2023
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Feb 09 2008
EXTENSIONS
References to A135504 added by Antti Karttunen, Mar 07 2023
STATUS
approved