A393999 Achilles numbers whose prime power factor exponents are pairwise coprime.

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1944, 2000, 2312, 2592, 2888, 3087, 3200, 3267, 3456, 3872, 3888, 4000, 4232, 4563, 4608, 5000, 5324, 5408, 5488, 6075, 6125, 6272, 6728, 6912, 7688, 7803, 8575

Offset

1,1

Comments

Proper subset of Achilles numbers A052486.

Superset of A393816 (Achilles numbers with exactly 2 distinct prime factors); a(78) = 21600 = 2^5 * 3^3 * 5^2.

For n > 1, A087315(n) = Product_{i=1..n} prime(i)^prime(n-1+1) is the smallest number in this sequence with n distinct prime factors.

Links

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

Index entries for sequences related to powerful numbers.

Formula

{a(n)} = A052486 \ A394000.

Example

Let s = A052486 and let t = A087315.
Table of n, a(n) for select n:
   n                     a(n)
-----------------------------------------------------------
   1       s(1) =        72 = 2^3 * 3^2              = t(2)
  14      s(14) =      1125 = 3^2 * 5^3
  20      s(21) =      1944 = 2^3 * 3^5
  46      s(53) =      8575 = 5^2 * 7^3
  78     s(100) =     21600 = 2^5 * 3^3 *  5^2       = t(3)
 102     s(137) =     36000 = 2^5 * 3^2 *  5^3
 110     s(151) =     42336 = 2^5 * 3^3 *  7^2
 119     s(165) =     48600 = 2^3 * 3^5 *  5^2
 169     s(251) =     98784 = 2^5 * 3^2 *  7^3
 174     s(258) =    104544 = 2^5 * 3^3 * 11^2
 293     s(490) =    324000 = 2^5 * 3^4 *  5^3
4631   s(14728) = 190512000 = 2^7 * 3^5 *  5^3 * 7^2 = t(4)

Mathematica

nn = 10^4; fQ[x_] := CoprimeQ @@ FactorInteger[x][[All, -1]]; Rest@ Union@ Flatten@ Table[If[fQ[#], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ]

Crossrefs

Cf. A001694, A052486, A087315, A126706, A286708, A393816, A394000.

Sequence in context: A052486 A393816 A393941 * A390952 A378859 A114128

Adjacent sequences: A393996 A393997 A393998 * A394000 A394001 A394002

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 06 2026

Status

approved