A392927 Achilles numbers k with at least 1 repeated exponent of prime power factors.

1800, 2700, 3528, 4500, 5292, 5400, 7200, 8712, 9000, 9800, 10584, 12168, 12348, 13068, 13500, 14112, 18252, 20808, 24200, 24300, 24500, 24696, 25992, 26136, 28800, 31212, 33075, 33800, 34300, 34848, 36504, 37044, 38088, 38988, 39200, 47432, 47628, 47916, 48672

Offset

1,1

Comments

Terms have more than 2 distinct prime factors, since we need at least one pair of prime power factors with the same exponent M > 1, and at least one prime power factor with an exponent larger than 1 and coprime to M.

Proper subset of A393708, in turn a proper subset of A375055; A393708(12) = A375055(2585) = 10800 = 2^4 * 3^3 * 5^2 has distinct prime power factor exponents and is not a term in this sequence.

Smallest a(n) with m distinct prime factors is 2*A002110(m)^2 for m > 2.

Links

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

Index entries for sequences related to powerful numbers

Example

Let s = A052486.
Table of n, a(n) for select n:
    n                     a(n)
-------------------------------------------------------------------
    1      s(21) =       1800 = 2^3 * 3^2 * 5^2
    2      s(26) =       2700 = 2^2 * 3^3 * 5^2
    3      s(32) =       3528 = 2^3 * 3^2 * 7^2
    4      s(37) =       4500 = 2^2 * 3^2 * 5^3
    5      s(41) =       5292 = 2^2 * 3^3 * 7^2
    7      s(51) =       7200 = 2^5 * 3^2 * 5^2
   27     s(130) =      33075 = 3^3 * 5^2 * 7^2
   60     s(238) =      88200 = 2^3 * 3^2 * 5^2 * 7^2
   68     s(266) =     108000 = 2^5 * 3^3 * 5^3
 1459    s(3269) =   10672200 = 2^3 * 3^2 * 5^2 * 7^2 * 11^2
28297   s(46748) = 1803601800 = 2^3 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2

Mathematica

nn = 50000; s = Union@ Flatten@ Rest@ Table[k = a^2*b^3; If[GCD @@ FactorInteger[k][[All, -1]] == 1, k, Nothing], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]; Select[s, ! DuplicateFreeQ[FactorInteger[#][[All, -1]] ] &]

Crossrefs

Cf. Supersets: A000977, A001694, A013929, A024619, A052486, A126706, A286708, A393708.

Sequence in context: A393708 A394000 A394050 * A179695 A375013 A380847

Adjacent sequences: A392924 A392925 A392926 * A392928 A392929 A392930

Keyword

nonn

Author

Michael De Vlieger, Mar 18 2026

Status

approved