A360013 Numbers whose exponent of 2 in their canonical prime factorization is larger than all the other exponents.

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 200, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256

Offset: 1

Author

Amiram Eldar, Jan 21 2023

Keywords

• nonn
• easy

Comments

Numbers k such that A007814(k) > A051903(A000265(k)).

The powers of 2 (A000079), except for 1, are all terms.

The product of any two terms (not necessarily distinct) is also a term.

This sequence is a disjoint union of {2} and the subsequences of numbers m of the form 2^k*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 4*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 8*o where o is an odd cubefree number; etc.

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/(zeta(k)*2*(2^k-1)) / Sum_{k>=2} 1/(zeta(k)*2*(2^k-1)) = 3.10346728882748723133... . [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.

Also even numbers whose multiset of prime factors has unique mode 2. - Gus Wiseman, Jul 10 2023

Examples

			From _Gus Wiseman_, Jul 09 2023: (Start)
108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is missing because 2 is not the unique mode.
120 = 2*2*2*3*5 is present because its unique mode is 2.
The terms together with their prime factorizations begin:
   2 = 2
   4 = 2*2
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  20 = 2*2*5
  24 = 2*2*2*3
  28 = 2*2*7
  32 = 2*2*2*2*2
  40 = 2*2*2*5
  44 = 2*2*11
  48 = 2*2*2*2*3
  52 = 2*2*13
  56 = 2*2*2*7
  60 = 2*2*3*5
  64 = 2*2*2*2*2*2
(End)
		

Links

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

Crossrefs

Cf. A000079, A000265, A007814, A051903, A056911.

Equals A360015 \ A360014.

Partitions of this type are counted by A241131.

Allowing any unique mode gives A356862, complement A362605.

Allowing any unique co-mode gives A359178, complement A362606.

Not requiring the mode to be unique gives A360015.

The opposite version is A362616, counted by A362612.

For co-mode instead of mode we have A364061, counted by A364062.

With least prime factor instead of 2, we have A364160, counted by A364193.

With a different factorization, we have the subsequence A335738.

A124010 gives prime signature, ordered A118914.

A362611 counts modes in prime factorization, triangle A362614.

A362613 counts co-modes in prime factorization, triangle A362615.

A363486 gives least mode in prime indices, A363487 greatest.

Cf. A001222, A002865, A327473, A327476, A362608, A362610, A363723, A363727.

Programs

Mathematica

q[n_] := Module[{e = IntegerExponent[n, 2], m}, m = n/2^e; (m == 1 && e > 0) || AllTrue[FactorInteger[m][[;; , 2]], # < e &]]; Select[Range[256], q]

PARI

is(n) = {my(e = valuation(n, 2), m = n >> e); (m == 1 && e > 0) || (m > 1 && vecmax(factor(m)[,2]) < e)};

Formula

a(n) = 2*A360015(n). - Gus Wiseman, Jul 10 2023