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 A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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, # == 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: A076702 A051761 A153326 * A168363 A182046 A171468 Adjacent sequences:  A338322 A338323 A338324 * A338326 A338327 A338328 KEYWORD nonn AUTHOR Amiram Eldar, Oct 22 2020 STATUS approved

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Last modified July 29 03:53 EDT 2021. Contains 346340 sequences. (Running on oeis4.)