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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).

%I A135506 #59 May 11 2023 16:56:52
%S A135506 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,
%T A135506 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,
%U A135506 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
%N 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).
%C A135506 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.
%C A135506 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
%C A135506 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
%C A135506 Per the prior comment, the distinct prime terms correspond to A045344. This is every prime except for 3. - _Bill McEachen_, Sep 12 2022
%H A135506 Antti Karttunen, <a href="/A135506/b135506.txt">Table of n, a(n) for n = 1..65537</a> (first 10000 terms from Robert Israel)
%H A135506 Eric S. Rowland, <a href="https://www.youtube.com/watch?v=OpaKpzMFOpg">Prime-Generating Recurrences and a Tale of Logarithmic Scale</a>, YouTube video, 2023. (See especially the last section beginning at 20:08).
%H A135506 Serafín Ruiz-Cabello, <a href="http://arxiv.org/abs/1504.05041">On the use of the lowest common multiple to build a prime-generating recurrence</a>, arXiv:1504.05041 [math.CO], 2015.
%F A135506 a(n) = A135504(n+1)/A135504(n) - 1.
%F A135506 a(n) = (n+1) / A361470(n). - _Antti Karttunen_, Mar 26 2023
%p A135506 x[1]:= 1;
%p A135506 for n from 2 to 101 do
%p A135506   x[n]:= x[n-1] + ilcm(x[n-1],n);
%p A135506   a[n-1]:= x[n]/x[n-1]-1;
%p A135506 od:
%p A135506 seq(a[n],n=1..100); # _Robert Israel_, Jan 11 2015
%t A135506 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 *)
%o A135506 (PARI) x1=1;for(n=2,40,x2=x1+lcm(x1,n);t=x1;x1=x2;print1(x2/t-1,","))
%o A135506 (Python)
%o A135506 from itertools import count, islice
%o A135506 from math import lcm
%o A135506 def A135506_gen(): # generator of terms
%o A135506     x = 1
%o A135506     for n in count(2):
%o A135506         y = x+lcm(x,n)
%o A135506         yield y//x-1
%o A135506         x = y
%o A135506 A135506_list = list(islice(A135506_gen(),20)) # _Chai Wah Wu_, Mar 13 2023
%Y A135506 Cf. A045344, A135504, A361460, A361461 (positions of 1's), A361462 [= a(n) mod 4], A361463, A361464, A361470.
%Y A135506 Cf. also A106108.
%K A135506 nonn
%O A135506 1,1
%A A135506 _Benoit Cloitre_, Feb 09 2008
%E A135506 References to A135504 added by _Antti Karttunen_, Mar 07 2023