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A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).

%I A362754 #13 May 09 2023 09:55:05
%S A362754 1,6,10,12,15,18,20,24,30,14,21,28,36,40,42,35,50,45,48,60,56,54,70,
%T A362754 63,66,55,22,33,44,72,75,78,65,26,39,52,84,80,90,98,96,100,102,85,34,
%U A362754 51,68,108,105,110,99,88,114,95,38,57,76,120,112,126,130,117,104,132,135,138,115,46,69,92,144
%N A362754 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and also contains as a factor the smallest prime that is not a factor of a(n-1).
%C A362754 No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A360519 and A361606. A close examination of the lines of concentrated terms, see the attached images, shows they have a slight downward curvature. In the first 250000 terms the only fixed points are 1, 69, 87, 116825, although it is possible more exist for very large values of n.
%H A362754 Scott R. Shannon, <a href="/A362754/b362754.txt">Table of n, a(n) for n = 1..10000</a>
%H A362754 Michael De Vlieger, <a href="/A362754/a362754.png">Log log scatterplot of a(n)</a>, n = 1..2^12, showing squarefree composites in green, numbers neither squarefree nor prime powers in blue, and numbers in A286708 in large light blue.
%H A362754 Scott R. Shannon, <a href="/A362754/a362754_1.png">Image of the first 250000 terms</a>. The green line is a(n) = n.
%H A362754 Scott R. Shannon, <a href="/A362754/a362754_2.png">Image of the first 250000 terms in color</a>. Terms with a lowest prime factor 2, 3, 5, 7, 11, >=13 are colored white, red, yellow, green, blue, violet and light gray respectively.
%e A362754 a(3) = 10 as a(2) = 6 = 2*3, and 10 is the smallest unused number that shares a factor with 6 while also containing 5 as a prime factor, the smallest prime not a factor of 6.
%e A362754 a(4) = 12 as a(3) = 10 = 2*5, and 12 is the smallest unused number that shares a factor with 10 while also containing 3 as a prime factor, the smallest prime not a factor of 10.
%t A362754 nn = 120; c[_] := False; m[_] := 2; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}]; j = a[2]; Do[q = 2; While[Divisible[j, q], q = NextPrime[q]]; k = m[q]; While[Or[c[#], PrimePowerQ[#], CoprimeQ[j, k]] &[q k], k++]; k *= q; While[c[m[q] q], m[q]++]; Set[{a[n], c[k], j}, {k, True, k}], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger_, May 02 2023 *)
%Y A362754 Cf. A337687, A351495, A064413, A360519, A361606.
%K A362754 nonn
%O A362754 1,2
%A A362754 _Scott R. Shannon_, May 02 2023