A338325 - OEIS

%I #10 Oct 23 2020 03:28:18

%S 1,4,8,9,25,27,36,49,72,100,108,121,125,169,196,200,216,225,289,343,

%T 361,392,441,484,500,529,675,676,841,900,961,968,1000,1089,1125,1156,

%U 1225,1323,1331,1352,1369,1372,1444,1521,1681,1764,1800,1849,2116,2197,2209

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

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

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

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

%H Amiram Eldar, <a href="/A338325/b338325.txt">Table of n, a(n) for n = 1..10000</a>

%H Massoud H. Dehkordi, <a href="https://hdl.handle.net/2134/12177">Asymptotic formulae for some arithmetic functions in number theory</a>, Ph.D. thesis, Loughborough University, 1998.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Biquadratefree.html">Biquadratefree</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.

%F 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).

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

%e 4 = 2^2 is a term since the exponent of its only prime factor is 2.

%e 72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

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

%Y Intersection of A001694 and A046100.

%Y Subsequences: A062503, A062838.

%Y Cf. A005117, A090699, A244000, A330595.

%K nonn

%O 1,2

%A _Amiram Eldar_, Oct 22 2020