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A376384 Numbers k such that there exists at least two m <= k such that both rad(m) | k and m is neither squarefree nor a prime power, i.e., m is in A126706, where rad = A007947.

%I A376384 #25 Oct 05 2024 10:58:08
%S A376384 18,24,30,36,40,42,48,50,54,56,60,66,70,72,75,78,80,84,88,90,96,98,
%T A376384 100,102,104,105,108,110,112,114,120,126,130,132,135,136,138,140,144,
%U A376384 147,150,152,154,156,160,162,165,168,170,174,176,180,182,184,186,189,190
%N A376384 Numbers k such that there exists at least two m <= k such that both rad(m) | k and m is neither squarefree nor a prime power, i.e., m is in A126706, where rad = A007947.
%C A376384 Numbers k such that A376505(k) > 1. A376505(k) >= 1 for all k in A126706.
%C A376384 Numbers k such that the cardinality of the intersection of row n of A162306 and A126706 exceeds 1.
%C A376384 Excludes prime powers; subsequence of A024619.
%C A376384 a(n) is not in A366825, since for k in A366825, there is only one m <= k that is in A126706, and that is k itself.
%H A376384 Michael De Vlieger, <a href="/A376384/b376384.txt">Table of n, a(n) for n = 1..10000</a>
%H A376384 Michael De Vlieger, <a href="/A376384/a376384.png">Hasse diagrams of row a(n) of A162306</a> for n = 1..12, showing numbers m in A126706 in blue, primes in red, perfect prime powers in gold, and squarefree composites in green.
%F A376384 a(n) = card({ m <= a(n) : rad(m) | a(n), Omega(m) > omega(m) > 1 }), where Omega = A001222 and omega = A001221.
%e A376384 Table showing the intersection of A126706 and row a(n) of A162306 for n = 1..12:
%e A376384 18: {12, 18},
%e A376384 24: {12, 18, 24},
%e A376384 30: {12, 18, 20, 24},
%e A376384 36: {12, 18, 24, 36},
%e A376384 40: {20, 40},
%e A376384 42: {12, 18, 24, 28, 36},
%e A376384 48: {12, 18, 24, 36, 48},
%e A376384 50: {20, 40, 50},
%e A376384 54: {12, 18, 24, 36, 48, 54},
%e A376384 56: {28, 56},
%e A376384 60: {12, 18, 20, 24, 36, 40, 45, 48, 50, 54, 60},
%e A376384 66: {12, 18, 24, 36, 44, 48, 54}.
%t A376384 Select[Range[2^8], Function[n, 1 < Count[Range[n], _?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], Nor[SquareFreeQ[#], PrimePowerQ[#]]] &)] ] ]
%Y A376384 Cf. A000961, A001221, A001222, A005117, A007947, A024619, A126706, A162306.
%K A376384 nonn,easy
%O A376384 1,1
%A A376384 _Michael De Vlieger_, Oct 02 2024