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

1,1

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

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

A176359 is a subsequence.

Sequence in context: A135590 A187859 A250137 * A111029 A124581 A177493

Adjacent sequences: A109396 A109397 A109398 * A109400 A109401 A109402

Keyword

nonn,easy,changed

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Extensions

Edited by Matthew Vandermast, Dec 07 2010

Status

approved