OFFSET
1,1
COMMENTS
Composite prime powers p^e such that e > 1 and p^e > 4 have the property j < k. With rad(p^e) = p, in the case of p = 2, pq = 6, 6 < 2^e for e > 2. In the case of odd p, we have 2p < p^e for e > 1.
Squarefree k do not have this property, since rad(k) = k, thus, kq > k by definition of prime q.
For k in this sequence, omega(j) > omega(k), but Omega(j) <= Omega(k), where omega(n) = A001221(n), and Omega(n) = A001222(n).
Subset of A126706.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, 1016 X 1016 pixel bitmap of n, n = 1..1032256, showing n in black if A126706(n) is in this sequence, else white.
EXAMPLE
k = 12 is not in the sequence since rad(k)*q(k) = 6*5 = 30, and 30 exceeds k. 18 and 24 are also not in the sequence for the same reason.
k = 36 is in the sequence since rad(36)*q(36) = 6*5 = 30, and 30 < 36.
k = 45 is in the sequence since rad(45)*q(45) = 15*2 = 30, and 30 < 45.
MATHEMATICA
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
q[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
Select[Select[Range[325], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], rad[#]*q[#] < # &] (* Michael De Vlieger, Mar 05 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Mar 05 2023
STATUS
approved