A386433 Achilles numbers with a primorial squarefree kernel that are not products of primorials.

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

Author

Michael De Vlieger, Jul 21 2025

Keywords

• nonn

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A001597, A001694, A002110, A007947, A025487, A055932, A056808, A377854, A378002.

Programs

Mathematica

(* 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[#]} &]

Formula

Let rad = A007947, omega = A001221, and P = A002110.

rad(a(n)) = P(omega(a(n))).

Intersection of A052486 and A056808 = A377854 \ A378002.