A394000 Achilles numbers whose prime power factor exponents are not all pairwise coprime.

1800, 2700, 3528, 4500, 5292, 5400, 7200, 8712, 9000, 9800, 10584, 10800, 12168, 12348, 13068, 13500, 14112, 16200, 18000, 18252, 20808, 21168, 24200, 24300, 24500, 24696, 25992, 26136, 28800, 31212, 31752, 33075, 33800, 34300, 34848, 36504, 37044, 38088, 38988

Offset

1,1

Comments

Proper subset of A393708 = A052486 intersect A000977, Achilles numbers with at least 3 distinct prime factors.

Smallest a(n) with m distinct prime factors is 2*A002110(m)^2 = 2*A061742(m) for m > 2.

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 \ A393999.

Example

Let s = A052486.
Table of n, a(n) for select n:
 n              a(n)
--------------------------------------------
 1    s(20) =  1800 = 2^3 * 3^2 *  5^2
 2    s(25) =  2700 = 2^2 * 3^3 *  5^2
 3    s(31) =  3528 = 2^3 * 3^2 *  7^2
 4    s(36) =  4500 = 2^2 * 3^2 *  5^3
 5    s(40) =  5292 = 2^2 * 3^3 *  7^2
 6    s(42) =  5400 = 2^3 * 3^3 *  5^2
 7    s(50) =  7200 = 2^5 * 3^2 *  5^2
 8    s(54) =  8712 = 2^3 * 3^2 * 11^2
 9    s(57) =  9000 = 2^3 * 3^2 *  5^3
12    s(64) = 10800 = 2^4 * 3^3 *  5^2
72   s(226) = 81675 = 3^3 * 5^2 * 11^2
76   s(237) = 88200 = 2^3 * 3^2 *  5^2 * 7^2

Mathematica

nn = 2^16; fQ[x_] := And[GCD @@ # == 1, ! CoprimeQ @@ #] &@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, A001694, A052486, A126706, A286708, A393708, A393999.

Sequence in context: A069476 A395197 A393708 * A394050 A392927 A179695

Adjacent sequences: A393997 A393998 A393999 * A394001 A394002 A394003

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 06 2026

Status

approved