A109399 Numbers with at least two 3s in their prime signature.

216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

• nonn
• easy

Comments

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.

Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).

The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024

Examples

			216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000578, A001235, A176297, A176313, A176350.

A176359 is a subsequence.

Programs

Mathematica

f[n_]:=Count[Last/@FactorInteger[n],3]>1; Select[Range[8!],f]

PARI

is(n)=#select(e->e==3, factor(n)[,2])>1 \\ Charles R Greathouse IV, Oct 19 2015

Extensions

Edited by Matthew Vandermast, Dec 07 2010