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%I #33 May 12 2023 05:47:13
%S 1,6,10,15,12,20,30,42,35,40,60,84,70,45,18,50,75,24,80,90,105,14,36,
%T 120,140,21,48,150,210,154,33,54,110,135,66,100,165,72,130,180,126,
%U 175,160,168,195,170,78,225,190,96,240,280,63,102,270,315,28,108,300,350,147,114,330,420,77,22
%N a(1) = 1, a(2) = 6, a(3) = 10; for n > 3, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and a(n-2) and also contains as factors the smallest primes that are not factors of both a(n-1) and a(n-2).
%C No term can be a prime power as each term must contain at least two distinct prime factors. This make the sequence similar to A362754, A360519 and A361606. Some small composite numbers take many terms to appear, e.g., a(354476) = 65. Such terms are usually preceded by a term that contains all the lower primes as factors. In the first 500000 terms, other than the first term, there are no fixed points, and it is unknown if any exist.
%H Michael De Vlieger, <a href="/A362600/a362600_2.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H Michael De Vlieger, <a href="/A362600/a362600_3.png">Log log scatterplot of a(n)</a>, n = 1..2^12, showing squarefree numbers in green and nonsquarefree numbers in blue, highlighting nonsquarefree numbers that are powerful (in A001694) with large light blue dots.
%H Scott R. Shannon, <a href="/A362600/a362600.png">Image of the first 250000 terms</a>. The green line is a(n) = n.
%H Scott R. Shannon, <a href="/A362600/a362600_1.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 a(4) = 15 as a(2) = 6 = 2*3 and a(3) = 10 = 2*5, and 15 is the smallest unused number that shares a factor with 6 and 10 while also containing 5 and 3 as prime factors, the smallest primes not factors of 6 and 10 respectively. This is the first term to differ from A362754.
%t nn = 120; c[_] := False;
%t f[x_] := If[OddQ[x], 2, y = 3; While[Divisible[x, y], y = NextPrime[y]]; y];
%t MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 6, 10}];
%t i = a[2]; j = a[3]; q = 5; u = 12;
%t Do[qq = f[j]; k = Ceiling[u/#] &[q*qq];
%t While[Or[c[#], CoprimeQ[i, #], CoprimeQ[i, j]] &[k*q*qq], k++];
%t k *= q*qq;
%t Set[{a[n], c[k], i, j, q}, {k, True, j, k, qq}];
%t If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, 4, nn}];
%t Array[a, nn] (* _Michael De Vlieger_, May 09 2023 *)
%Y Cf. A362754, A337687, A351495, A064413, A360519, A361606, A053669.
%K nonn
%O 1,2
%A _Scott R. Shannon_, May 02 2023
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