OFFSET
1,2
COMMENTS
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.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in green, and other numbers in blue.
Michael De Vlieger, Plot p(i)^e(i) | a(n) at (x,y) = (n,i) 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.
EXAMPLE
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.
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.
MATHEMATICA
nn = 120; c[_] := False; m[_] := 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3];
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jun 02 2023
STATUS
approved