A394050 Achilles numbers with more than 2 distinct prime factors, whose exponents that are all pairwise indivisible.

1800, 2700, 3528, 4500, 5292, 5400, 7200, 8712, 9000, 9800, 10584, 12168, 12348, 13068, 13500, 14112, 18252, 20808, 21600, 24200, 24300, 24500, 24696, 25992, 26136, 28800, 31212, 33075, 33800, 34300, 34848, 36000, 36504, 37044, 38088, 38988, 39200, 42336, 47432

Offset

1,1

Comments

Numbers k in A393708 (intersection of A000977 and A052486) that are also in A317101.

All Achilles numbers in A393816 (with just 2 distinct prime factors) have prime exponents that pairwise indivisible.

It is not always true that a(n) is either in A392927 or in A394224; a(4128739) = 31492800000000 = 2^12 * 3^9 * 5^8 is in neither.

Links

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

Example

Let s = A052486, t = A392927, and q = A394224.
Table of n, a(n) for select n:
        n                               a(n)
  --------------------------------------------------------------------------
        1        s(20) =    t(1) =     1800 =  2^3 * 3^2 * 5^2
        2        s(25) =    t(2) =     2700 =  2^2 * 3^3 * 5^2
        3        s(31) =    t(3) =     3528 =  2^3 * 3^2 * 7^2
        4        s(36) =    t(4) =     4500 =  2^2 * 3^2 * 5^3
        5        s(40) =    t(5) =     5292 =  2^2 * 3^3 * 7^2
       19       s(100) =    q(1) =    21600 =  2^5 * 3^3 * 5^2
       28       s(129) =   t(27) =    33075 =  3^3 * 5^2 * 7^2
       32       s(137) =    q(2) =    36000 =  2^5 * 3^2 * 5^3
       38       s(151) =    q(3) =    42336 =  2^5 * 3^3 * 7^2
       65       s(237) =   t(60) =    88200 =  2^3 * 3^2 * 5^2 * 7^2
     1592      s(3268) = t(1459) = 10672200 =  2^3 * 3^2 * 5^2 * 7^2 * 11^2
  4128739   s(6505162) =     31492800000000 = 2^12 * 3^9 * 5^8
.
A393708(12) = 10800 = 2^4 * 3^3 * 5^2 is not in this sequence since among exponents, we have 2 | 4.

Mathematica

nn = 2^16; fQ[x_] := And[Length[#] > 2, GCD @@ # == 1, Count[Tuples[#, 2], _?(And[UnsameQ @@ #, Divisible @@ #] &)] == 0] &[FactorInteger[x][[All, -1]] ]; Union@ Flatten@ Table[If[fQ[#], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ]

Crossrefs

Cf. A000977, A052486, A317101, A392927, A393708, A393816, A394000, A394224.

Sequence in context: A395197 A393708 A394000 * A392927 A179695 A375013

Adjacent sequences: A394047 A394048 A394049 * A394051 A394052 A394053

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 06 2026

Status

approved