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A360014
Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximum of the other exponents.
6
1, 6, 10, 14, 22, 26, 30, 34, 36, 38, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 94, 100, 102, 106, 110, 114, 118, 122, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 180, 182, 186, 190, 194, 196, 202, 206, 210, 214, 216, 218, 222, 226, 230, 238, 246, 252
OFFSET
1,2
COMMENTS
Numbers k such that A007814(k) = A051903(A000265(k)).
This sequence is a disjoint union of {1}, the even squarefree numbers (A039956), and the subsequences of even k-free numbers that are not (k-1)-free, for k >= 3. These subsequences include, for k = 3, numbers of the form 4*o where o is an odd cubefree number that is not squarefree (i.e., an odd term of A067259).
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 0.222707226888193809... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-2)/(zeta(k)*2*(2^k-1)) = 0.24575013985660328894... .
This sequence is a subsequence of A360015 and the asymptotic density of this sequence within A360015 is exactly 1/2.
LINKS
MATHEMATICA
q[n_] := 2^(e = IntegerExponent[n, 2]) < n && e == Max[FactorInteger[n/2^e][[;; , 2]]]; q[1] = True; Select[Range[250], q]
PROG
(PARI) is(n) = {my(e = valuation(n, 2), m = n >> e); n == 1 ||(m > 1 && e == vecmax(factor(m)[, 2]))};
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 21 2023
STATUS
approved