A382062 Powerful numbers whose number of divisors is divisible by their number of unitary divisors.

1, 8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 343, 392, 432, 500, 512, 648, 675, 864, 968, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1728, 1944, 2000, 2048, 2187, 2197, 2312, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 4000, 4232, 4563, 4913, 5000, 5324, 5400

Offset: 1

Author

Amiram Eldar, Mar 14 2025

Keywords

• nonn
• easy

Comments

Powerful numbers k such that A034444(k) | A000005(k).

The primitive terms of A382061: if k is a term and m is a squarefree number that is coprime to k, then k*m is a term of A382061. The asymptotic density of A382061 can be calculated using the terms of this sequence (see A382061 for a formula).

Examples

			27 = 3^3 is a term since it is powerful, A000005(27) = 4, A034444(27) = 2, and 2 | 4.
72 = 2^3 * 3^2 is a term since it is powerful, A000005(72) = 12, A034444(72) = 4, and 4 | 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Intersection of A001694 and A382061.

Cf. A000005, A034444, A382064.

Programs

Mathematica

q[k_] := Module[{e = FactorInteger[k][[;;, 2]]}, AllTrue[e, # > 1 &] && Divisible[Times @@ (e+1), 2^Length[e]]]; Select[Range[5400], # == 1 || q[#] &]

PARI

isok(k) = if(k == 1, 1, my(f = factor(k)); vecmin(f[,2]) > 1 && !(numdiv(f) % (1<