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A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3. 11
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
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.
Sequence in context: A051761 A153326 A368959 * A168363 A182046 A171468
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 22 2020
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)