A360015 Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.

1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 100, 102, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 128, 130, 132, 134, 136, 138

Offset: 1

Author

Amiram Eldar, Jan 21 2023

Keywords

• nonn
• easy

Comments

Numbers k such that A007814(k) = A051903(k).

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)) / Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 2.10346728882748723133... . [corrected by Amiram Eldar, Jul 10 2025]

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

Examples

			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)
		

Links

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

Crossrefs

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

Partitions of this type are counted by A241131.

The case of unique mode is A360013, complement here A360014.

For unique minimal prime exponent we have A364061, counted by A364062.

For minimal prime exponent we have A364158, counted by A364159.

A027746 lists prime factors (with multiplicity).

A112798 lists prime indices, length A001222, sum A056239.

A362611 counts modes in prime indices, triangle A362614.

Cf. A002865, A327473, A327476, A356862, A359178, A362605, A362613, A363486.

Programs

Mathematica

q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]

PARI

is(n) = n == 1 || vecmax(factor(n)[,2]) == valuation(n, 2);

Formula

Disjoint union of A360013 and A360014.

a(n) = A360013(n)/2. - Gus Wiseman, Jul 14 2023