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

Author

Amiram Eldar, Jan 21 2023

Keywords

• nonn
• easy

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)) / Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 1.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]

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

Crossrefs

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

Equals A360015 \ A360013.

Programs

Mathematica

q[n_] := 2^(e = IntegerExponent[n, 2]) < n && e == Max[FactorInteger[n/2^e][[;; , 2]]]; q[1] = True; Select[Range[250], q]

PARI

is(n) = {my(e = valuation(n, 2), m = n >> e); n == 1 ||(m > 1 && e == vecmax(factor(m)[,2]))};