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A368103 a(1)=1; for n>1, a(n) is the smallest number not already used which has a factor difference in common with a(n-1).

%I A368103 #52 Jan 06 2025 22:05:52
%S A368103 1,4,9,16,7,27,40,10,18,8,3,15,24,6,2,12,5,21,32,45,13,28,54,26,42,20,
%T A368103 30,14,36,17,57,80,35,48,23,75,11,39,56,72,22,46,94,144,19,63,88,43,
%U A368103 135,55,91,112,25,49,64,31,99,120,38,60,29,93,128,33,65,84,41,129,176,50,66,92,141,192
%N A368103 a(1)=1; for n>1, a(n) is the smallest number not already used which has a factor difference in common with a(n-1).
%C A368103 A factor difference of x is any abs(p-q) where x=p*q (in other words, the difference of a factor pair of x, per A368312).
%C A368103 Prime numbers are among the numbers which appear most delayed in this sequence. - _Thomas Scheuerle_, Dec 12 2023
%H A368103 Neal Gersh Tolunsky, <a href="/A368103/b368103.txt">Table of n, a(n) for n = 1..10000</a>
%H A368103 Thomas Scheuerle, <a href="/A368103/a368103_1.m.txt">MATLAB Script</a>.
%H A368103 Thomas Scheuerle, <a href="/A368103/a368103.png">Plot of the first 393 prime numbers over their indices of appearance</a>. It is remarkable that prime numbers do not appear in order.
%e A368103 For n=2, a(1)=1 can be factored only as 1*1, which has difference 0. The next term cannot be 2 and 3 as they do not have a factor difference 0, but 4 = 2*2 does, so that a(2) = 4.
%e A368103 For n=5, a(4)=16 has factor differences 0,6,15 and the smallest unused number with one of those differences is a(5) = 7 = 7*1 difference 6.
%o A368103 (MATLAB) % See Scheuerle link.
%Y A368103 Cf. A368312.
%Y A368103 Cf. A368059 (with factor sums), A359035, A360995.
%K A368103 nonn
%O A368103 1,2
%A A368103 _Neal Gersh Tolunsky_, Dec 11 2023