A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

1, 4, 8, 9, 25, 27, 36, 49, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 289, 343, 361, 392, 441, 484, 500, 529, 675, 676, 841, 900, 961, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1681, 1764, 1800, 1849, 2116, 2197, 2209

Offset

1,2

Comments

Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.

Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.

Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".

Links

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

Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Eric Weisstein's World of Mathematics, Biquadratefree.

Eric Weisstein's World of Mathematics, Powerful Number.

Formula

The number of terms not exceeding x is asymptotically (zeta(3/2)/zeta(3)) * J_2(1/2) * x^(1/2) + (zeta(2/3)/zeta(2)) * J_2(1/3) * x^(1/3), where J_2(s) = Product_{p prime} (1 - p^(-4*s) - p^(-5*s) - p^(-6*s) + p^(-7*s) + p^(-8*s)) (Dehkordi, 1998).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).

Example

4 = 2^2 is a term since the exponent of its only prime factor is 2.
72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

Mathematica

Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]

Crossrefs

Intersection of A001694 and A046100.

Subsequences: A062503, A062838.

Cf. A005117, A090699, A244000, A330595.

Sequence in context: A051761 A153326 A368959 * A168363 A182046 A171468

Adjacent sequences: A338322 A338323 A338324 * A338326 A338327 A338328

Keyword

nonn

Author

Amiram Eldar, Oct 22 2020

Status

approved