A360014 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximum of the other exponents.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A000265, A007814, A039956, A051903, A067259.

Equals A360015 \ A360013.

Sequence in context: A315232 A175168 A315233 * A315234 A315235 A080784

Adjacent sequences: A360011 A360012 A360013 * A360015 A360016 A360017

Keyword

nonn,easy

Author

Amiram Eldar, Jan 21 2023

Status

approved