cp's OEIS Frontend

A362600 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).

%I A362600 #33 May 12 2023 05:47:13
%S A362600 1,6,10,15,12,20,30,42,35,40,60,84,70,45,18,50,75,24,80,90,105,14,36,
%T A362600 120,140,21,48,150,210,154,33,54,110,135,66,100,165,72,130,180,126,
%U A362600 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 A362600 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 A362600 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 A362600 Michael De Vlieger, <a href="/A362600/a362600_2.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%H A362600 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 A362600 Scott R. Shannon, <a href="/A362600/a362600.png">Image of the first 250000 terms</a>. The green line is a(n) = n.
%H A362600 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 A362600 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 A362600 nn = 120; c[_] := False;
%t A362600 f[x_] := If[OddQ[x], 2, y = 3; While[Divisible[x, y], y = NextPrime[y]]; y];
%t A362600 MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 6, 10}];
%t A362600 i = a[2]; j = a[3]; q = 5; u = 12;
%t A362600 Do[qq = f[j]; k = Ceiling[u/#] &[q*qq];
%t A362600   While[Or[c[#], CoprimeQ[i, #], CoprimeQ[i, j]] &[k*q*qq], k++];
%t A362600   k *= q*qq;
%t A362600   Set[{a[n], c[k], i, j, q}, {k, True, j, k, qq}];
%t A362600   If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, 4, nn}];
%t A362600 Array[a, nn] (* _Michael De Vlieger_, May 09 2023 *)
%Y A362600 Cf. A362754, A337687, A351495, A064413, A360519, A361606, A053669.
%K A362600 nonn
%O A362600 1,2
%A A362600 _Scott R. Shannon_, May 02 2023