cp's OEIS Frontend

A363444 a(n) = n for n <= 3; for n > 3, a(n) is the smallest positive number that has not yet appeared that includes as factors the distinct prime factors of a(n-2) and a(n-1) that are not shared between a(n-2) and a(n-1).

%I A363444 #16 Jun 07 2023 00:08:36
%S A363444 1,2,3,6,4,9,12,8,15,30,10,18,45,20,24,60,5,36,90,25,42,210,35,48,420,
%T A363444 70,21,120,140,63,150,280,84,75,350,126,105,40,168,315,50,252,525,80,
%U A363444 294,630,55,462,840,110,231,1050,220,693,1260,330,77,1470,660,154,735,990,308,945,1320,616,1155
%N A363444 a(n) = n for n <= 3; for n > 3, a(n) is the smallest positive number that has not yet appeared that includes as factors the distinct prime factors of a(n-2) and a(n-1) that are not shared between a(n-2) and a(n-1).
%C A363444 The last prime to appear in the first 10000 terms is a(17) = 5, and it is unknown if more appear. The largest terms increase rapidly in size, e.g., a(8924) = 2233642178577810, although subsequent terms can be significantly smaller. It is unknown is all numbers eventually appear.
%H A363444 Michael De Vlieger, <a href="/A363444/b363444.txt">Table of n, a(n) for n = 1..10000</a>
%H A363444 Michael De Vlieger, <a href="/A363444/a363444.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A363444 Michael De Vlieger, <a href="/A363444/a363444_1.png">Log log scatterplot of a(n)</a>, n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in green, and other numbers in blue.
%H A363444 Michael De Vlieger, <a href="/A363444/a363444_2.png">Plot p(i)^e(i) | a(n) at (x,y) = (n,i)</a> for n = 1..2048, 12X vertical exaggeration, with a color function representing e(i), where black indicates e(i) = 1, red indicates e(i) = 2, yellow-green = 3, green = 4, and blue = 5. The bar at bottom indicates primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
%e A363444 a(4) = 6 as a(2) = 2 and a(3) = 3 contain the distinct prime factors 2 and 3 respectively, both of which only appear in one term. Therefore a(4) is the smallest unused number that contains both 2 and 3 as factors, which is 6.
%e A363444 a(6) = 9 as a(4) = 6 = 2*3 and a(5) = 4 = 2*2, so 3 is the only prime factor that is not shared between these terms. Therefore a(6) is the smallest unused number that contains 3 as a factor, which is 9.
%t A363444 nn = 120; c[_] := False; m[_] := 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3];
%t A363444 i = {a[2]}; j = {a[3]}; Do[q = Times @@ SymmetricDifference[i, j]; While[c[Set[k, q m[q]]], m[q]++]; Set[{a[n], c[k], i, j}, {k, True, j, FactorInteger[k][[All, 1]]}], {n, 4, nn}]; Array[a, nn] (* _Michael De Vlieger_, Jun 05 2023 *)
%Y A363444 Cf. A098550, A336957, A362600, A362754, A351495, A359804.
%K A363444 nonn
%O A363444 1,2
%A A363444 _Scott R. Shannon_, Jun 02 2023