A381313 Numbers that are divisible by the cube of an odd prime.

27, 54, 81, 108, 125, 135, 162, 189, 216, 243, 250, 270, 297, 324, 343, 351, 375, 378, 405, 432, 459, 486, 500, 513, 540, 567, 594, 621, 625, 648, 675, 686, 702, 729, 750, 756, 783, 810, 837, 864, 875, 891, 918, 945, 972, 999, 1000, 1026, 1029, 1053, 1080, 1107, 1125

Offset: 1

Author

Amiram Eldar, Feb 19 2025

Keywords

• nonn
• easy

Comments

Numbers whose odd part is not cubefree.

The asymptotic density of this sequence is 1 - 8/(7*zeta(3)) = 1 - 1/A233091 = 0.04924871705062003579... .

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A046099 and A038838.

Cf. A000265, A002117, A233091.

Programs

Mathematica

cubeFreeQ[k_] := Max[FactorInteger[k][[;;, 2]]] < 3; q[k_] := !cubeFreeQ[k / 2^IntegerExponent[k, 2]]; Select[Range[1200], q]

PARI

iscubefree(k) = if(k == 1, 1, vecmax(factor(k)[, 2]) < 3);
isok(k) = !iscubefree(k >> valuation(k, 2));