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

1,2

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).

Links

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

Index entries for sequences related to powerful numbers.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Intersection of A001694 and A382061.

Cf. A000005, A034444, A382064.

Sequence in context: A262675 A102834 A376171 * A370786 A377820 A116002

Adjacent sequences: A382059 A382060 A382061 * A382063 A382064 A382065

Keyword

nonn,easy

Author

Amiram Eldar, Mar 14 2025

Status

approved