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A366952 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with n but does not equal n, and shares a factor with a(n-1).

%I A366952 #13 Oct 30 2023 09:40:48
%S A366952 1,4,6,2,10,8,14,12,3,15,33,9,39,18,20,22,34,16,38,24,27,30,46,26,40,
%T A366952 28,21,7,203,35,155,50,36,32,42,44,74,48,45,5,205,60,86,52,54,56,94,
%U A366952 58,70,25,75,65,265,80,66,62,72,64,118,68,122,76,57,78,13,104,134,82,84,49,497,63,219
%N A366952 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with n but does not equal n, and shares a factor with a(n-1).
%C A366952 The sequence is conjectured to be a permutation of the positive integers, although the primes typically take many terms to appear, e.g., a(95890) = 223. When a prime does appear it is often followed by a term that is significantly larger than the average-sized term. See the examples below. The primes do not occur in their natural order.
%H A366952 Scott R. Shannon, <a href="/A366952/b366952.txt">Table of n, a(n) for n = 1..10000</a>
%H A366952 Scott R. Shannon, <a href="/A366952/a366952.png">Image of the first 100000 terms with a(n) < 300000</a>. The green line is a(n) = n.
%e A366952 a(3) = 6 as 6 does not equal 3, shares the factor 3 with 3 while sharing the factor 2 with a(2) = 4.
%e A366952 a(29) = 203 as 203 does not equal 29, shares the factor 29 with 29 while sharing the factor 7 with a(28) = 7. This is an example of both n and a(n-1) being primes which forces a(n) to be significantly larger than the average-sized term.
%Y A366952 Cf. A366908, A159193, A093714, A085229, A119018, A000027.
%K A366952 nonn
%O A366952 1,2
%A A366952 _Scott R. Shannon_, Oct 29 2023