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A364934 a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.

%I A364934 #35 Aug 19 2023 00:13:36
%S A364934 2,2,3,2,4,5,2,6,8,8,9,4,10,12,14,13,2,14,15,8,16,17,2,18,22,20,23,2,
%T A364934 22,23,3,8,24,29,2,26,28,28,29,3,10,32,31,2,32,33,13,4,34,36,42,43,2,
%U A364934 38,39,17,4,40,43,3,15,21,21,22,43,4,43,5,9,19,3,20,48,58,48,60,63,28,52,53,2,51,28
%N A364934 a(n+1) = 1 + number of previous terms that share a factor > 1 with a(n); a(1) = 2.
%C A364934 There are prominent lines that have more terms, their coefficients are approximately: 0.519, 0.329, 0.689, 0.188, 0.615, ... (see the frequency link). They seem to be distorted prime harmonic lines: 1/2, 1/3, 2/3, 1/5, 3/5, ... from A016035.
%C A364934 It appears limsup a(n)/n is approximately 0.83.
%H A364934 Rok Cestnik, <a href="/A364934/b364934.txt">Table of n, a(n) for n = 1..1000</a>
%H A364934 Rok Cestnik, <a href="/A364934/a364934.pdf">a(n)/n frequency for 3*10^5 terms</a>.
%H A364934 Michael De Vlieger, <a href="/A364934/a364934.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%e A364934 [2,*]          1 term  shares a factor with 2, so a(2) = 1+1 = 2.
%e A364934 [2,2,*]        2 terms share  a factor with 2, so a(3) = 1+2 = 3.
%e A364934 [2,2,3,*]      1 term  shares a factor with 3, so a(4) = 1+1 = 2.
%e A364934 [2,2,3,2,*]    3 terms share  a factor with 2, so a(5) = 1+3 = 4.
%e A364934 [2,2,3,2,4,*]  4 terms share  a factor with 4, so a(6) = 1+4 = 5.
%t A364934 nn = 120; c[_] := 0; s = {}; a[1] = j = 2; Do[c[j]++; If[c[j] == 1, AppendTo[s, j]]; k = 1 + Total@ Map[c[#] Boole[GCD[#, j] > 1] &, s]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Aug 16 2023 *)
%o A364934 (Python)
%o A364934 from sympy.ntheory import primefactors
%o A364934 a=[2]; p = [{2}];
%o A364934 for n in range(1,1000):
%o A364934     a.append(sum(1 if p[-1].intersection(p[i]) else 0 for i in range(n))+1)
%o A364934     p.append(set(primefactors(a[-1])))
%o A364934 (Python)
%o A364934 from math import gcd, prod
%o A364934 from sympy import factorint
%o A364934 from itertools import islice
%o A364934 from collections import Counter
%o A364934 def agen(): # generator of terms
%o A364934     a=[2]; f=2; c=Counter([f])
%o A364934     while True:
%o A364934         yield a[-1]
%o A364934         a.append(1 + sum(c[fi] for fi in c if gcd(f,fi)>1))
%o A364934         f=prod(factorint(a[-1]).keys())
%o A364934         c[f] += 1
%o A364934 a=list(islice(agen(), 1000)) # _Michael S. Branicky_, Aug 16 2023
%Y A364934 Cf. A016035, A124056.
%K A364934 nonn
%O A364934 1,1
%A A364934 _Rok Cestnik_, Aug 15 2023