OFFSET
1,2
COMMENTS
In other words, a(n) is the smallest novel k satisfying the above constraint on omega(k) such that the squarefree kernel of i*j*k is a primorial number. Conjectured to be a permutation of the positive integers (A000027), with primes in order.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..5002
EXAMPLE
a(1,2) = 1,2 and a(3) = 3 is least k such that rad(1*2*3) = 6 is primorial, with omega(1,2,3) = 0,1,1 satisfied.
a(2,3) = 2,3 with omega(2,3) = 1,1 so a(4) cannot be 4 (even though rad(2*3*4) = 6), because omega(4) = 1. Therefore a(4) = 6, the least k satisfying the omega condition (omega(6) = 2) such that rad(2*3*6) = 6 is primorial.
a(3,4) = 3,6 with omega(3,6) = 1,2 therefore a(5) = 4 since rad(3*6*4) = 6 and no smaller term is available at this point (omega constraint is not invoked).
MATHEMATICA
nn = 120; c[_] := False;
f[x_] := Or[IntegerQ@ Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi@ FactorInteger[x][[All, 1]] == {1}]];
Array[Set[{a[#], c[#]}, {#, True}] &, 3];
Set[{i, j, u, v, w}, {2, 3, 4, 1, 1}];
Do[m = i*j; k = u;
While[Or[c[k], Equal @@ {v, w, Set[x, PrimeNu[k]]}, ! f[m*k]], k++];
Set[{a[n], c[k], i, j, v, w}, {k, True, j, k, w, x}];
If[k == u, While[c[u], u++]], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 03 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Jun 01 2024
EXTENSIONS
More terms from Michael De Vlieger, Jun 03 2024
STATUS
approved