A360014 - OEIS

%I #9 Jan 23 2023 02:33:02

%S 1,6,10,14,22,26,30,34,36,38,42,46,58,62,66,70,74,78,82,86,94,100,102,

%T 106,110,114,118,122,130,134,138,142,146,154,158,166,170,174,178,180,

%U 182,186,190,194,196,202,206,210,214,216,218,222,226,230,238,246,252

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

%C Numbers k such that A007814(k) = A051903(A000265(k)).

%C 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).

%C The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 0.222707226888193809... .

%C 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... .

%C This sequence is a subsequence of A360015 and the asymptotic density of this sequence within A360015 is exactly 1/2.

%H Amiram Eldar, <a href="/A360014/b360014.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%Y Equals A360015 \ A360013.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Jan 21 2023