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A364997 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) > k but rad(k)*A053669(k) < k.

%I A364997 #10 Jul 20 2024 20:05:21
%S A364997 40,45,56,63,88,99,104,117,136,152,153,171,175,176,184,207,208,232,
%T A364997 248,261,272,275,279,280,296,297,304,315,325,328,333,344,351,368,369,
%U A364997 376,387,423,424,425,440,459,464,472,475,477,488,495,496,513,520,531,536,539
%N A364997 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) > k but rad(k)*A053669(k) < k.
%C A364997 Subset of A126706, numbers that are neither squarefree nor prime powers.
%C A364997 For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
%C A364997 A355432(k) = 0, A360543(k) > 0. There exist m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k, but nondivisors m < k do not exist such that rad(m) = rad(k).
%H A364997 Michael De Vlieger, <a href="/A364997/b364997.txt">Table of n, a(n) for n = 1..10000</a>
%H A364997 Michael De Vlieger, <a href="/A364997/a364997.png">Annotated plot of b(n) = A126706(n)</a>, n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus for n = 1..400. Terms in this sequence are colored black, those in A364999 in blue, in A364998 in gold, and in A361098 in red.
%H A364997 Michael De Vlieger, <a href="/A364997/a364997_1.png">Plot of b(n)</a>, n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus for n = 1..14400 using the same color scheme as immediately above.
%H A364997 Michael De Vlieger, <a href="/A364997/a364997_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 b(n) in this sequence are colored black, else white.
%F A364997 Intersection of A360765 and A360767.
%e A364997 Let b(n) = A126706(n), S = A360767, and T = A360765.
%e A364997 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 in S but not in T.
%e A364997 b(2) = 18 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, neither is less than 18, hence 18 is not in S but is in T.
%e A364997 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 not in S but is in T.
%e A364997 b(7) = a(1) = 40 since p*r = 5*10 = 50 and q*r = 3*10 = 30. We have both 50 > 40 and 30 < 40, thus 40 is in both S and T, etc.
%t A364997 Select[Select[Range[500], 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 A364997 Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360765, A360767, A361098, A364998, A364999.
%K A364997 nonn
%O A364997 1,1
%A A364997 _Michael De Vlieger_, Aug 16 2023