A378002 Achilles numbers that are products of primorials.

72, 288, 432, 864, 1152, 1800, 2592, 3456, 4608, 5400, 6912, 7200, 10368, 10800, 15552, 18432, 21600, 27648, 28800, 31104, 41472, 43200, 54000, 55296, 62208, 64800, 73728, 86400, 88200, 93312, 108000, 115200, 124416, 162000, 165888, 172800, 194400, 221184, 259200

Offset: 1

Author

Michael De Vlieger, Nov 16 2024

Keywords

• nonn
• easy

Comments

Products of primorials that are powerful but not perfect powers.

Examples

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  288 = 2^5 * 3^2
   a(3) =  432 = 2^4 * 3^3
   a(4) =  864 = 2^5 * 3^3
   a(5) = 1152 = 2^7 * 3^2
   a(6) = 1800 = 2^3 * 3^2 * 5^2
   a(7) = 2592 = 2^5 * 3^4
   a(8) = 3456 = 2^7 * 3^3
   a(9) = 4608 = 2^9 * 3^2
  a(10) = 5400 = 2^3 * 3^3 * 5^2
  a(11) = 6912 = 2^8 * 3^3
  a(12) = 7200 = 2^5 * 3^2 * 5^2
		

Links

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

Crossrefs

Cf. A001597, A001694, A002110, A007947, A052486, A025487, A286708, A364930, A377854.

Programs

Mathematica

(* First load function f in A025487, then: *)
Select[Rest@ Union@ Flatten@ f[14],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A286708 \ A001597 and A025487.

Intersection of A052486 and A025487.

Proper subset of A364930, in turn a proper subset of A369374.

Proper subset of A377854.