A382064 - OEIS

A382064

Cubefull numbers whose number of coreful divisors is divisible by their number of exponential divisors.

1, 256, 432, 512, 648, 2000, 4096, 5000, 5184, 5488, 6561, 6912, 10125, 11664, 16875, 19208, 19683, 21296, 27783, 32000, 35152, 40000, 41472, 52488, 54000, 62208, 64827, 78608, 81000, 87808, 107811, 109744, 110592, 117128, 135000, 148176, 153664, 177957, 186624

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Cubefull numbers k such that A049419(k) | A005361(k).

The primitive terms of A382063: if k is a term and m is a cubefree number that is coprime to k, then k*m is a term of A382063.

The asymptotic density of A382063 can be calculated using the terms of this sequence (see A382063 for a formula).

LINKS

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

Index entries for sequences related to powerful numbers.

EXAMPLE

256 = 2^8 is a term since it is cubefull, A005361(256) = 8, A049419(256) = 4, and 4 | 8.

432 = 2^4 * 3^3 is a term since it is cubefull, A005361(432) = 12, A049419(432) = 6, and 6 | 12.

MATHEMATICA

q[k_] := Module[{e = FactorInteger[k][[;; , 2]]}, AllTrue[e, # > 2 &] && Divisible[Times @@ e, Times @@ DivisorSigma[0, e]]]; Select[Range[140000], # == 1 || q[#] &]

PROG

(PARI) isok(k) = if(k == 1, 1, my(e = factor(k)[, 2]); vecmin(e) > 2 && !(vecprod(e) % vecprod(apply(x -> numdiv(x), e))));

CROSSREFS

Intersection of A036966 and A382063.

Cf. A004709, A005361, A049419, A382062.

Sequence in context: A115176 A299156 A221259 * A223693 A223064 A206206

Adjacent sequences: A382061 A382062 A382063 * A382065 A382066 A382067

KEYWORD

nonn

AUTHOR

Amiram Eldar, Mar 14 2025

STATUS

approved