A377854 Achilles numbers whose squarefree kernel is a primorial.

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

Author

Michael De Vlieger, Nov 16 2024

Keywords

• nonn
• easy

Comments

Numbers whose squarefree kernel is a primorial that are powerful but not a perfect power.

Examples

			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
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Fast Mathematica algorithm for A055932 (function f).

Crossrefs

Cf. A001597, A001694, A002110, A007947, A052486, A055932, A286708, A369374, A378002.

Programs

Mathematica

(* 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[#]} &]

Formula

Intersection of A286708 \ A001597 and A055932.

Intersection of A052486 and A055932.

Proper subset of A369374.

Superset of A378002.