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A377854 Achilles numbers whose squarefree kernel is a primorial.

%I A377854 #45 Nov 17 2024 07:09:36
%S A377854 72,108,288,432,648,864,972,1152,1800,1944,2592,2700,3456,3888,4500,
%T A377854 4608,5400,6912,7200,8748,9000,10368,10800,13500,15552,16200,17496,
%U A377854 18000,18432,21600,23328,24300,27648,28800,31104,34992,36000,40500,41472,43200,45000,48600
%N A377854 Achilles numbers whose squarefree kernel is a primorial.
%C A377854 Numbers whose squarefree kernel is a primorial that are powerful but not a perfect power.
%H A377854 Michael De Vlieger, <a href="/A377854/b377854.txt">Table of n, a(n) for n = 1..10000</a>
%H A377854 Michael De Vlieger, <a href="/A377854/a377854.txt">Fast Mathematica algorithm for A055932</a> (function f).
%F A377854 Intersection of A286708 \ A001597 and A055932.
%F A377854 Intersection of A052486 and A055932.
%F A377854 Proper subset of A369374.
%F A377854 Superset of A378002.
%e A377854 Prime power decomposition of the first 12 terms:
%e A377854    a(1) =   72 = 2^3 * 3^2
%e A377854    a(2) =  108 = 2^2 * 3^3
%e A377854    a(3) =  288 = 2^5 * 3^2
%e A377854    a(4) =  432 = 2^4 * 3^3
%e A377854    a(5) =  648 = 2^3 * 3^4
%e A377854    a(6) =  864 = 2^5 * 3^3
%e A377854    a(7) =  972 = 2^2 * 3^5
%e A377854    a(8) = 1152 = 2^7 * 3^2
%e A377854    a(9) = 1800 = 2^3 * 3^2 * 5^2
%e A377854   a(10) = 1944 = 2^3 * 3^5
%e A377854   a(11) = 2592 = 2^5 * 3^4
%e A377854   a(12) = 2700 = 2^2 * 3^3 * 5^2
%t A377854 (* First load function f in the link, then: *)
%t A377854 Select[Rest@ Union@ Flatten@ f[10],
%t A377854  And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
%t A377854    GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]
%Y A377854 Cf. A001597, A001694, A002110, A007947, A052486, A055932, A286708, A369374, A378002.
%K A377854 nonn,easy
%O A377854 1,1
%A A377854 _Michael De Vlieger_, Nov 16 2024