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

Author

Amiram Eldar, Oct 22 2020

Keywords

• nonn

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".

Examples

			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.
		

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.

Crossrefs

Intersection of A001694 and A046100.

Subsequences: A062503, A062838.

Cf. A005117, A090699, A244000, A330595.

Programs

Mathematica

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

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).