A335738 Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is a power of 2.

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 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, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256, 260, 264, 268, 272

Offset: 1

Author

Peter Munn, Jun 20 2020

Keywords

• nonn

Comments

2 is the only term not divisible by 4. All powers of 2 are present. Every term divisible by an odd square is divisible by 16, the first such being 144.

The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.

Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k-1)(m), is 2. k can be shown to be A299090(m).

Closed under squaring, but not closed under multiplication: 12 = 3^1 * 2^2 and 432 = 3^1 * 3^2 * 2^4 are in the sequence, but 12 * 432 = 5184 = 3^4 * 2^6 = 2^2 * 6^4 is not.

The asymptotic density of this sequence is Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.21363357193921052068..., where d(k) = 2^(k-1)/((2^k-1)*zeta(k)) is the asymptotic density of odd k-free numbers for k >= 2, and d(1) = 0. - Amiram Eldar, Feb 10 2024

Examples

			6 is a squarefree number, so its factorization for the definition (into powers of nonunit squarefree numbers with distinct exponents that are powers of 2) is the trivial "6^1". 6^1 is therefore the factor with the largest exponent, and is not a power of 2, so 6 is not in the sequence.
48 factorizes for the definition as 3^1 * 2^4. The factor with the largest exponent is 2^4, which is a power of 2, so 48 is in the sequence.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. The factor with the largest exponent, 10^64, is a power of 10, not a power of 2, so 10^100 is not in the sequence.
		

Links

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

Crossrefs

Complement within A020725 of A335740.

A000188, A299090 are used in a formula defining this sequence.

Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).

Subsequences: A000079\{1}, A001749, A181818\{1}, A273798.

Numbers in the even bisection of A336322.

Row m of A352780 essentially gives the defined factorization of m.

Programs

Mathematica

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 300], FixedPointList[s, #] [[-3]] == 2 &] (* Amiram Eldar, Nov 27 2020 *)

PARI

is(n) = {my(e = valuation(n, 2), o = n >> e); if(e == 0, 0, if(o == 1, n > 1, floor(logint(e, 2)) > floor(logint(vecmax(factor(o)[,2]), 2))));} \\ Amiram Eldar, Feb 10 2024

Formula

{a(n)} = {m : m >= 2 and A000188^(k-1)(m) = 2, where k = A299090(m)}.

{a(n)} = {m : m >= 2 and A352780(m,e) = 2^(2^e), where e = A299090(m)-1}. - Peter Munn, Jun 24 2022