A393708 Achilles numbers with more than 2 distinct prime factors.

1800, 2700, 3528, 4500, 5292, 5400, 7200, 8712, 9000, 9800, 10584, 10800, 12168, 12348, 13068, 13500, 14112, 16200, 18000, 18252, 20808, 21168, 21600, 24200, 24300, 24500, 24696, 25992, 26136, 28800, 31212, 31752, 33075, 33800, 34300, 34848, 36000, 36504, 37044

Offset

1,1

Comments

Intersection of A000977 and A052486.

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(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
10   s(60)  =  9800 = 2^3 * 5^2 *  7^2
33   s(129) = 33075 = 3^3 * 5^2 *  7^2
81   s(237) = 88200 = 2^3 * 3^2 *  5^2 * 7^2

Maple

filter:= proc(n) local F;
   F:= ifactors(n)[2];
   nops(F) > 2 and andmap(`>`, F[.., 2], 1) and igcd(F[.., 2])=1
end proc:
select(F, [$1..50000]); # Robert Israel, Mar 03 2026

Mathematica

With[{nn = 40000}, Union@ Flatten@ Rest@ Table[If[And[Length[#] > 2, GCD @@ # == 1] &@ FactorInteger[#][[All, -1]], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

Prog

(PARI) isok(k)=my(f=factor(k)); k>9 && vecmin(f[, 2])>1 && gcd(f[, 2])==1 && omega(f)>2; \\ Michel Marcus, Mar 03 2026

Crossrefs

Cf. A000977, A001694, A013929, A024619, A052486, A126706, A286708, A375055, A390950.

Sequence in context: A035891 A069476 A395197 * A394000 A394050 A392927

Adjacent sequences: A393705 A393706 A393707 * A393709 A393710 A393711

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 02 2026

Status

approved