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A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).

%I A380473 #37 Aug 14 2025 03:21:19
%S A380473 126,168,198,234,264,306,312,342,408,414,456,522,552,558,666,696,738,
%T A380473 744,774,846,888,954,984,990,1032,1062,1098,1128,1170,1206,1272,1278,
%U A380473 1314,1320,1386,1416,1422,1464,1494,1530,1560,1602,1608,1638,1650,1704,1710,1746
%N A380473 Numbers k neither squarefree nor prime power (i.e., in A126706) such that A119288(k) <= A003557(k) < A053669(k) < A006530(k).
%C A380473 Let rad = A007947, p = A119288, q = A053669, g = A006530, and r = A003557.
%C A380473 Numbers k in A126706 such that p <= r < q < g.
%C A380473 Terms are products k of a number s in A033845 and a number t in A007310 with at least one prime power factor p^m | k such that m > 1.
%H A380473 Michael De Vlieger, <a href="/A380473/b380473.txt">Table of n, a(n) for n = 1..10000</a>
%F A380473 Intersection of A364998 and A080259 = A364998 \ A055932 = A364998 \ A369540.
%e A380473 Table of n, a(n) for select n:
%e A380473    n    a(n)                       r   q
%e A380473   --------------------------------------
%e A380473    1    126 = 2 * 3^2 * 7          3   5
%e A380473    2    168 = 2^3 * 3 * 7          4   5
%e A380473    3    198 = 2 * 3^2 * 11         3   5
%e A380473    4    234 = 2 * 3^2 * 13         3   5
%e A380473    5    264 = 2^3 * 3 * 11         4   5
%e A380473    6    306 = 2 * 3^2 * 17         3   5
%e A380473    7    312 = 2^3 * 3 * 13         4   5
%e A380473   24    990 = 2 * 3^2 * 5 * 11     3   7
%e A380473   29   1170 = 2 * 3^2 * 5 * 13     3   7
%e A380473   45   1650 = 2 * 3 * 5^2 * 11     5   7
%e A380473   57   1980 = 2^2 * 3^2 * 5 * 11   6   7
%e A380473   68   2340 = 2^2 * 3^2 * 5 * 13   6   7
%t A380473 a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ]; q];
%t A380473 s = Select[Range[2^12], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
%t A380473 Select[s, And[#3 < #4 < #2[[-1, 1]], #2[[2, 1]] <= #3] & @@
%t A380473   {#1, #2, #1/Apply[Times, #2[[All, 1]]], a053669[#1]} & @@
%t A380473   {#, FactorInteger[#]} &]
%Y A380473 Cf. A003557, A007310, A007947, A033845, A053669, A055932, A080259, A119288, A126706, A364998, A369540.
%K A380473 nonn
%O A380473 1,1
%A A380473 _Michael De Vlieger_, Jul 22 2025