cp's OEIS Frontend

A360765 Numbers k that are neither prime powers nor squarefree, such that A007947(k) * A053669(k) < k.

%I A360765 #9 Mar 07 2023 02:32:48
%S A360765 36,40,45,48,50,54,56,63,72,75,80,88,96,98,99,100,104,108,112,117,135,
%T A360765 136,144,147,152,153,160,162,171,175,176,184,189,192,196,200,207,208,
%U A360765 216,224,225,232,240,242,245,248,250,252,261,270,272,275,279,280,288,294,296,297,300,304,315,320,324,325
%N A360765 Numbers k that are neither prime powers nor squarefree, such that A007947(k) * A053669(k) < k.
%C A360765 Let rad(k) = A007947(k), and let q = A053669(k).
%C A360765 Let j = A007947(k)*A053669(k) = rad(k)*q.
%C A360765 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.
%C A360765 Squarefree k do not have this property, since rad(k) = k, thus, kq > k by definition of prime q.
%C A360765 For k in this sequence, omega(j) > omega(k), but Omega(j) <= Omega(k), where omega(n) = A001221(n), and Omega(n) = A001222(n).
%C A360765 Subset of A126706.
%H A360765 Michael De Vlieger, <a href="/A360765/b360765.txt">Table of n, a(n) for n = 1..10000</a>
%H A360765 Michael De Vlieger, <a href="/A360765/a360765.png">1016 X 1016 pixel bitmap of n</a>, n = 1..1032256, showing n in black if A126706(n) is in this sequence, else white.
%e A360765 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.
%e A360765 k = 36 is in the sequence since rad(36)*q(36) = 6*5 = 30, and 30 < 36.
%e A360765 k = 45 is in the sequence since rad(45)*q(45) = 15*2 = 30, and 30 < 45.
%t A360765 rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
%t A360765 q[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
%t A360765 Select[Select[Range[325], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], rad[#]*q[#] < # &] (* _Michael De Vlieger_, Mar 05 2023 *)
%Y A360765 Cf. A001221, A001222, A007947, A053669, A126706, A246547.
%K A360765 nonn
%O A360765 1,1
%A A360765 _Michael De Vlieger_, Mar 05 2023