A392134 Achilles numbers that are neither 4-free nor 4-full.

288, 432, 648, 800, 864, 972, 1152, 1568, 1944, 2000, 3200, 3456, 3872, 4000, 4608, 5000, 5408, 5488, 6075, 6272, 6912, 7200, 8748, 9248, 10125, 10800, 10976, 11552, 11907, 12500, 12800, 14112, 15488, 16000, 16200, 16875, 16928, 17496, 18000, 18432, 19208, 21168

Offset

1,1

Comments

Intersection of A052486 (Achilles numbers) and A391115 (numbers that are neither 4-free nor 4-full).

A052486 is the union of this sequence, A375073 (noncubefull Achilles numbers), and A391011 (4-full Achilles numbers).

This sequence intersects both A388293 (cubefull Achilles numbers) and A390539 (noncubefull Achilles numbers).

Numbers whose set of prime power factor exponents m is setwise coprime, contains at least one m >= 4, but not all m >= 4, yet all m > 1.

Links

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

Index entries for sequences related to powerful numbers.

Example

Table of n, a(n) for select n:
 n    a(n)
---------------------------
 1    288 = 2^5 * 3^2
 2    432 = 2^4 * 3^3
 3    648 = 2^3 * 3^4
 4    800 = 2^5 * 5^2
 5    864 = 2^5 * 3^3
 6    972 = 2^2 * 3^5
 7   1152 = 2^7 * 3^2
 8   1568 = 2^5 * 7^2
 9   1944 = 2^3 * 3^5
10   2000 = 2^4 * 5^3
19   6075 = 3^5 * 5^2
22   7200 = 2^5 * 3^2 * 5^2

Mathematica

With[{nn = 25000}, Union@ Flatten@ Table[If[And[GCD @@ # == 1, 0 < Count[#, _?(# > 3 &)] < Length[#]] &[FactorInteger[#][[;; , -1]]], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]} ] ]

Prog

(PARI) is_A392134(n) = if(n<=1, 0, (e->(1==gcd(e) && 3==bitor(vecmin(e), 1) && vecmax(e)>3))(factor(n)[, 2])); \\ Antti Karttunen, Jan 22 2026

Crossrefs

Cf. A375073, A388293, A390539, A391011.

Supersets: A001694, A013929, A024619, A046101, A052486, A126706, A286708, A391115.

Sequence in context: A345807 A380937 A244196 * A280515 A061831 A250774

Adjacent sequences: A392131 A392132 A392133 * A392135 A392136 A392137

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 15 2026

Status

approved