A377854
Achilles numbers whose squarefree kernel is a primorial.
Original entry on oeis.org
72, 108, 288, 432, 648, 864, 972, 1152, 1800, 1944, 2592, 2700, 3456, 3888, 4500, 4608, 5400, 6912, 7200, 8748, 9000, 10368, 10800, 13500, 15552, 16200, 17496, 18000, 18432, 21600, 23328, 24300, 27648, 28800, 31104, 34992, 36000, 40500, 41472, 43200, 45000, 48600
Offset: 1
Prime power decomposition of the first 12 terms:
a(1) = 72 = 2^3 * 3^2
a(2) = 108 = 2^2 * 3^3
a(3) = 288 = 2^5 * 3^2
a(4) = 432 = 2^4 * 3^3
a(5) = 648 = 2^3 * 3^4
a(6) = 864 = 2^5 * 3^3
a(7) = 972 = 2^2 * 3^5
a(8) = 1152 = 2^7 * 3^2
a(9) = 1800 = 2^3 * 3^2 * 5^2
a(10) = 1944 = 2^3 * 3^5
a(11) = 2592 = 2^5 * 3^4
a(12) = 2700 = 2^2 * 3^3 * 5^2
-
(* First load function f in the link, then: *)
Select[Rest@ Union@ Flatten@ f[10],
And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]
A386433
Achilles numbers with a primorial squarefree kernel that are not products of primorials.
Original entry on oeis.org
108, 648, 972, 1944, 2700, 3888, 4500, 8748, 9000, 13500, 16200, 17496, 18000, 23328, 24300, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 78732, 81000, 97200, 112500, 121500, 132300, 135000, 139968, 144000, 145800, 180000, 209952, 218700, 220500
Offset: 1
Table of n, a(n), and A053669(a(n)) for n = 1..12.
n a(n) A053669(a(n))
--------------------------------------------
1 108 = 2^2 * 3^3 5
2 648 = 2^3 * 3^4 5
3 972 = 2^2 * 3^5 5
4 1944 = 2^3 * 3^5 5
5 2700 = 2^2 * 3^3 * 5^2 7
6 3888 = 2^4 * 3^5 5
7 4500 = 2^2 * 3^2 * 5^3 7
8 8748 = 2^2 * 3^7 5
9 9000 = 2^3 * 3^2 * 5^3 7
10 13500 = 2^2 * 3^3 * 5^3 7
11 16200 = 2^3 * 3^4 * 5^2 7
12 17496 = 2^3 * 3^7 5
Let s = A052486.
The number 12 is not a term since it is not powerful (i.e., not in A001694).
The number 36, though powerful, is not a term since it is a perfect square.
s(1) = 72 is not in this sequence since rad(72) = P(2) = 6 and 72 = 2*6*6 = P(1)*P(2)*P(2).
s(2) = 108 = 3*6*6 is in the sequence since it is not a product of primorials.
The number 144, though powerful, is not a term because it is a perfect square.
s(3) = 200 is not a term because rad(200) = 10 = 2*5 is not also divisible by A053669(200) = 3.
s(4) = 288 is not in this sequence since rad(288) = P(2) = 6 and 288 = 2*2*2*6*6 = P(1)*P(1)*P(1)*P(2)*P(2), etc.
-
(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
Select[Union@ Flatten[f[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[;; , -1]] == 1, Max@ Differences[#2[[All, -1]] ] > 0] & @@ {#, FactorInteger[#]} &]
Showing 1-2 of 2 results.
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