OFFSET
1,2
COMMENTS
The powers of 2 (A000079) are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {1} and the subsequences of numbers m of the form 2^(k-1)*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 2*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 4*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 0.44541445377638761933... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-1)/(zeta(k)*(2^k-1)) = 0.93691473348959419722... .
Also numbers whose multiset of prime factors has low (i.e. least) mode 2. Here, a mode in a multiset is an element that appears at least as many times as each of the others; for example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}. - Gus Wiseman, Jul 14 2023
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A360013(n)/2. - Gus Wiseman, Jul 14 2023
EXAMPLE
From Gus Wiseman, Jul 14 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is present because it has low mode 2.
The terms together with their prime factorizations begin:
1 =
2 = 2
4 = 2*2
6 = 2*3
8 = 2*2*2
10 = 2*5
12 = 2*2*3
14 = 2*7
16 = 2*2*2*2
20 = 2*2*5
22 = 2*11
24 = 2*2*2*3
26 = 2*13
28 = 2*2*7
30 = 2*3*5
32 = 2*2*2*2*2
34 = 2*17
36 = 2*2*3*3
(End)
MATHEMATICA
q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]
PROG
(PARI) is(n) = n == 1 || vecmax(factor(n)[, 2]) == valuation(n, 2);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 21 2023
STATUS
approved