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A378002 Achilles numbers that are products of primorials.

%I A378002 #26 Nov 17 2024 07:10:10
%S A378002 72,288,432,864,1152,1800,2592,3456,4608,5400,6912,7200,10368,10800,
%T A378002 15552,18432,21600,27648,28800,31104,41472,43200,54000,55296,62208,
%U A378002 64800,73728,86400,88200,93312,108000,115200,124416,162000,165888,172800,194400,221184,259200
%N A378002 Achilles numbers that are products of primorials.
%C A378002 Products of primorials that are powerful but not perfect powers.
%H A378002 Michael De Vlieger, <a href="/A378002/b378002.txt">Table of n, a(n) for n = 1..10000</a>
%F A378002 Intersection of A286708 \ A001597 and A025487.
%F A378002 Intersection of A052486 and A025487.
%F A378002 Proper subset of A364930, in turn a proper subset of A369374.
%F A378002 Proper subset of A377854.
%e A378002 Prime power decomposition of the first 12 terms:
%e A378002    a(1) =   72 = 2^3 * 3^2
%e A378002    a(2) =  288 = 2^5 * 3^2
%e A378002    a(3) =  432 = 2^4 * 3^3
%e A378002    a(4) =  864 = 2^5 * 3^3
%e A378002    a(5) = 1152 = 2^7 * 3^2
%e A378002    a(6) = 1800 = 2^3 * 3^2 * 5^2
%e A378002    a(7) = 2592 = 2^5 * 3^4
%e A378002    a(8) = 3456 = 2^7 * 3^3
%e A378002    a(9) = 4608 = 2^9 * 3^2
%e A378002   a(10) = 5400 = 2^3 * 3^3 * 5^2
%e A378002   a(11) = 6912 = 2^8 * 3^3
%e A378002   a(12) = 7200 = 2^5 * 3^2 * 5^2
%t A378002 (* First load function f in A025487, then: *)
%t A378002 Select[Rest@ Union@ Flatten@ f[14],
%t A378002  And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
%t A378002    GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]
%Y A378002 Cf. A001597, A001694, A002110, A007947, A052486, A025487, A286708, A364930, A377854.
%K A378002 nonn,easy
%O A378002 1,1
%A A378002 _Michael De Vlieger_, Nov 16 2024