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Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.
7

%I #10 Jul 20 2024 20:09:41

%S 18,24,90,120,126,150,168,180,198,234,264,306,312,342,408,414,456,522,

%T 552,558,630,666,696,738,744,774,840,846,888,954,984,990,1032,1050,

%U 1062,1098,1128,1170,1206,1260,1272,1278,1314,1320,1386,1416,1422,1464,1470,1494

%N Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.

%C Subset of A126706, numbers that are neither squarefree nor prime powers.

%C For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).

%C A355432(k) > 0, A360543(k) = 0. There exist nondivisors m < k such that rad(m) = rad(k); however, m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k do not exist.

%H Michael De Vlieger, <a href="/A364998/b364998.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A364998/a364998.png">Annotated plot of b(n) = A126706(n)</a>, with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364999 in blue, in A364997 in green, and in A361098 in red.

%H Michael De Vlieger, <a href="/A364998/a364998_1.png">Plot of b(n)</a>, with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms, using the same color scheme as described immediately above.

%H Michael De Vlieger, <a href="/A364998/a364998_2.png">Plot of b(n)</a>, with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates a strong quasiperiodic pattern approximately mod 169.

%F Intersection of A363082 and A360768.

%e Let b(n) = A126706(n), S = A360768, and T = A363082.

%e b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is not in S.

%e b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and T.

%e b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is in S but not T.

%e b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, thus 40 is neither in S nor T, etc.

%t Select[Select[Range[1500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r <= k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

%Y Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360768, A361098, A363082, A364997, A364999.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Aug 16 2023