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A381139 a(1) = 1, a(2) = 2. Let j = a(n-1) and let d = A160995(j) be the smallest non-divisor of j which shares a prime factor with j. Then for n > 2 a(n) is the smallest multiple of d which is not yet in the sequence.

%I A381139 #19 Apr 04 2025 22:34:06
%S A381139 1,2,4,6,8,12,16,18,20,24,9,30,28,36,32,42,40,48,27,54,44,60,56,66,52,
%T A381139 72,10,64,78,68,84,80,90,76,96,45,102,88,108,104,114,92,120,63,126,
%U A381139 100,132,112,138,116,144,50,124,150,128,156,136,162,140,168,81,174
%N A381139 a(1) = 1, a(2) = 2. Let j = a(n-1) and let d = A160995(j) be the smallest non-divisor of j which shares a prime factor with j. Then for n > 2 a(n) is the smallest multiple of d which is not yet in the sequence.
%C A381139 2 is the only prime in the sequence. Initially every odd term > 1 is a multiple of 9.
%C A381139 The first 3 odd terms not divisible by 9 are the following: a(19241) = 25, a(38481) = 75, a(57719) = 125, a(76959) = 175, a(115438) = 275. Differences between indices of odd terms divisible by 25 but not 9 are approximately 19240.
%C A381139 In this sequence, 25 follows 55440 = 2^4 * 3^2 * 5 * 7 * 11. The number 49 is missing for n <= 2^24.
%C A381139 Though a(n+1) must neither divide a(n) nor be coprime to same, a(n) may divide a(n+1). Examples: the sequence begins with {1, 2, 4}, a(19) = 27 and a(20) = 54, a(44) = 63 and a(45) = 126, etc.
%H A381139 Michael De Vlieger, <a href="/A381139/b381139.txt">Table of n, a(n) for n = 1..10000</a>
%H A381139 Michael De Vlieger, <a href="/A381139/a381139.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A381139 Michael De Vlieger, <a href="/A381139/a381139.txt">Prime power decomposition of a(n)</a>, n = 1..1000, (2025).
%e A381139 a(3) = 4, the smallest non divisor of 2 which shares a divisor with 2.
%e A381139 a(4) = 6, the smallest non divisor of 4 which shares a divisor with 4.
%e A381139 a(5) = 8 because 4 is the smallest non divisor of 6 with a factor in common but 4 has been seen earlier and 8 is the least novel multiple of 4.
%e A381139 a(10) = 24 implies a(11) = 9 since A160995(24) = 9, and 9 has not occurred earlier. This is the first odd number in the sequence. Thereafter we see a(19) = 27, a(36) = 45, a(44) = 63, a(61) = 81, etc (consecutive odd multiples of 9). It is not known if this behavior continues as the sequence extends.
%t A381139 nn = 120; c[_] := False; m[_] := 1; j = 2;
%t A381139 {1, 2}~Join~
%t A381139   Reap[Do[k = 2; Until[Nor[Divisible[j, k], CoprimeQ[j, k]], k++];
%t A381139     While[Or[c[#], Divisible[j, #], CoprimeQ[j, #]] &[k*m[k]], m[k]++];
%t A381139       k *= m[k]; c[k] = True; j = Sow[k],
%t A381139 {n, nn}] ][[-1, 1]] (* _Michael De Vlieger_, Feb 18 2025 *)
%Y A381139 Cf. A160995, A337687.
%K A381139 nonn,easy
%O A381139 1,2
%A A381139 _David James Sycamore_ and _Michael De Vlieger_, Feb 15 2025