A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

1, 16, 81, 384, 625, 640, 896, 1296, 1408, 1664, 2176, 2401, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, 6784, 7552, 7808, 8576, 9088, 9216, 9344, 10000, 10112, 10624, 10935, 11392, 12416, 12928, 13184, 13696, 13952, 14464, 14641, 15309, 16256, 16768

Offset: 1

Author

Amiram Eldar, Feb 25 2023

Keywords

• nonn

Comments

Numbers k such that A073184(k) = A190867(k).

Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).

If k and m are coprime terms, then k*m is also a term.

The characteristic function of this sequence depends only on prime signature.

1 is the only cubefree (A004709) term.

Includes the 4th powers of squarefree numbers (1 and A113849).

The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).

Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.

The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .

Links

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

Crossrefs

Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.

Subsequences: A030514, A113849, A179664, A360907.

Programs

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]

PARI

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }