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

%I A338325 #11 Feb 16 2025 08:34:00
%S A338325 1,4,8,9,25,27,36,49,72,100,108,121,125,169,196,200,216,225,289,343,
%T A338325 361,392,441,484,500,529,675,676,841,900,961,968,1000,1089,1125,1156,
%U A338325 1225,1323,1331,1352,1369,1372,1444,1521,1681,1764,1800,1849,2116,2197,2209
%N A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.
%C A338325 Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.
%C A338325 Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.
%C A338325 Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".
%H A338325 Amiram Eldar, <a href="/A338325/b338325.txt">Table of n, a(n) for n = 1..10000</a>
%H A338325 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 A338325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Biquadratefree.html">Biquadratefree</a>.
%H A338325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.
%F A338325 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 A338325 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).
%e A338325 4 = 2^2 is a term since the exponent of its only prime factor is 2.
%e A338325 72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.
%t A338325 Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]
%Y A338325 Intersection of A001694 and A046100.
%Y A338325 Subsequences: A062503, A062838.
%Y A338325 Cf. A005117, A090699, A244000, A330595.
%K A338325 nonn
%O A338325 1,2
%A A338325 _Amiram Eldar_, Oct 22 2020