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

1,1

Comments

Products of primorials that are powerful but not perfect powers.

Links

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

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.

Example

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

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[#]} &]

Crossrefs

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

Sequence in context: A316800 A004007 A279272 * A173546 A308136 A242534

Adjacent sequences: A377999 A378000 A378001 * A378003 A378004 A378005

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 16 2024

Status

approved