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

Offset: 1

Author

Amiram Eldar, Mar 14 2025

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

Intersection of A036966 and A382063.

Cf. A004709, A005361, A049419, A382062.

Programs

Mathematica

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

PARI

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