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A356432 a(1) = 1; for n > 1, a(n) is the smallest positive number not occurring earlier that shares a factor with a(n-1) plus the number of divisors of a(n-1).

%I A356432 #13 Jan 16 2023 14:56:50
%S A356432 1,2,4,7,3,5,14,6,8,9,10,12,15,19,18,16,21,20,13,24,22,26,25,28,17,38,
%T A356432 27,31,11,39,43,30,32,34,36,33,37,42,35,45,48,29,62,40,44,46,50,49,52,
%U A356432 54,56,58,60,51,55,59,61,57,122,63,23,65,66,64,71,73,69,146,68,70,72,74,75,78,76,41
%N A356432 a(1) = 1; for n > 1, a(n) is the smallest positive number not occurring earlier that shares a factor with a(n-1) plus the number of divisors of a(n-1).
%C A356432 The sequence is conjectured to be a permutation of the positive integers. In the first 250000 terms there are twenty-three fixed points: 1, 2, 12, 16, 27 ..., 2279, 5401, 7339. It is possibly no more exist although this is unknown.
%H A356432 Robert Israel, <a href="/A356432/b356432.txt">Table of n, a(n) for n = 1..10000</a>
%H A356432 Scott R. Shannon, <a href="/A356432/a356432.png">Image of the first 250000 terms</a>. The green line is y = n.
%e A356432 a(7) = 14 as a(6) = 5 which has two divisors, and 14 is the smallest unused number that shares a factor with 5 + 2 = 7.
%p A356432 A[1]:= 1; S:= {$2..5000}:
%p A356432 for i from 2 do
%p A356432   found:= false;
%p A356432   t:= A[i-1] + numtheory:-tau(A[i-1]);
%p A356432   for s in S do
%p A356432     if igcd(s,t) > 1 then
%p A356432       A[i]:= s;
%p A356432       found:= true;
%p A356432       S:= S minus {s};
%p A356432       break
%p A356432     fi
%p A356432   od;
%p A356432   if not found then break fi;
%p A356432 od:
%p A356432 seq(A[j],j=1..i-1); # _Robert Israel_, Jan 16 2023
%Y A356432 Cf. A356430, A356431, A000005, A354960, A348086.
%K A356432 nonn
%O A356432 1,2
%A A356432 _Scott R. Shannon_, Aug 07 2022