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A376834 Numbers k that have at least 1 powerful number m such that 1 < m <= k that are not prime powers such that rad(m) | k, where rad = A007947.

%I A376834 #5 Oct 13 2024 11:22:10
%S A376834 36,42,48,54,60,66,72,78,84,90,96,100,102,108,110,114,120,126,130,132,
%T A376834 138,140,144,150,156,160,162,168,170,174,180,186,190,192,196,198,200,
%U A376834 204,210,216,220,222,224,225,228,230,234,238,240,246,250,252,255,258,260
%N A376834 Numbers k that have at least 1 powerful number m such that 1 < m <= k that are not prime powers such that rad(m) | k, where rad = A007947.
%C A376834 Numbers k such that A286708 and row k of A162306 meet.
%C A376834 Contains A286708, since for k in A286708, m = k is such that rad(m) | k.
%H A376834 Michael De Vlieger, <a href="/A376834/b376834.txt">Table of n, a(n) for n = 1..10000</a>
%H A376834 Michael De Vlieger, <a href="/A376834/a376834.png">Plot p^e | a(n) at (x,y) = (n,pi(p))</a>, n = 1..1500, pi = A000720, with a color function indicating exponent e = 1 in black, e = 2 = red, e = 3 = orange, ..., maximum e in magenta. The indicator bar below the image represents squarefree a(n) in green, a(n) in A332785 in blue, and a(n) in A286708 in violet.
%e A376834 Table showing select values of a(n):
%e A376834    n    a(n)                A286708 Intersect row a(n) of A162306.
%e A376834   ---------------------------------------------------------------
%e A376834    1    36 = 2^2 * 3^2      {36}
%e A376834    2    42 = 2 * 3 * 7      {36}
%e A376834    3    48 = 2^4 * 3        {36}
%e A376834    4    54 = 2 * 3^3        {36}
%e A376834    5    60 = 2^2 * 3 * 5    {36}
%e A376834    6    66 = 2 * 3 * 11     {36}
%e A376834    7    72 = 2^3 * 3^2      {36, 72}
%e A376834    8    78 = 2 * 3 * 13     {36, 72}
%e A376834    9    84 = 2^2 * 3 * 7    {36, 72}
%e A376834   14   108 = 2^2 * 3^3      {36, 72, 108}
%e A376834   17   120 = 2^3 * 3 * 5    {36, 72, 100, 108}
%e A376834   24   150 = 2 * 3 * 5^2    {36, 72, 100, 108, 144}
%t A376834 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A376834 Reap[Do[m = 1; k = 0;
%t A376834   While[Nor[k == 2, m == n + 1],
%t A376834     If[And[Divisible[n, #], Divisible[m, #^2], ! PrimePowerQ[m] ] &[
%t A376834       rad[m]], k++]; m++];
%t A376834     If[k == 2, Sow[n]], {n, 2^10}] ][[-1, 1]]
%Y A376834 Cf. A001694, A007947, A162306, A286708.
%K A376834 nonn
%O A376834 1,1
%A A376834 _Michael De Vlieger_, Oct 06 2024