A335740 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 not a power of 2.

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90

Offset

1,1

Comments

Every missing number greater than 2 is a multiple of 4. Every power of 2 is missing. Every positive power of every squarefree number greater than 2 is present.

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 greater than 2. k can be shown to be A299090(m).

The asymptotic density of this sequence is 1 - Sum_{k>=0} (d(2^(k+1)) - d(2^k))/2^(2^(k+1)-1) = 0.78636642806078947931..., 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

Links

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

Formula

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

Example

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 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 not 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 in the sequence.

Mathematica

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 100], FixedPointList[s, #] [[-3]] > 2 &] (* Amiram Eldar, Nov 27 2020 *)

Prog

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

Crossrefs

Complement within A020725 of A335738.

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

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

Subsequences: A042968\{1,2}, A182853, A268390.

With {1}, numbers in the odd bisection of A336322.

Sequence in context: A256543 A186145 A364058 * A352873 A047984 A288513

Adjacent sequences: A335737 A335738 A335739 * A335741 A335742 A335743

Keyword

nonn

Author

Peter Munn, Jun 20 2020

Status

approved