A297404 A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal.

0, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1, 8, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 4, 3, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 32, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 9, 8, 1, 1, 3, 1, 1

Offset: 1

Author

Rémy Sigrist, Dec 29 2017

Keywords

• nonn
• easy
• base

Comments

This sequence is similar to A087207; here we encode the exponents, there the prime numbers appearing in the prime factorization of a number.

The binary representation of a(n) shows which exponents appear in the prime factorization of n, but without multiplicities:

- for any prime number p and k > 0, if p^k divides n but p^(k+1) does not divide n, then a(n) AND 2^(k-1) = 2^(k-1) (where AND denotes the bitwise AND operator),

- conversely, if a(n) AND 2^(k-1) = 2^(k-1) for some k > 0, then there is prime number p such that p^k divides n but p^(k+1) does not divide n.

Examples

			For n = 90:
- 90 = 5^1 * 3^2 * 2^1,
- the exponents appearing in the prime factorization of 90 are 1 and 2,
- hence a(90) = 2^(1-1) + 2^(2-1) = 3.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000120, A000695, A001694, A004709, A005117, A038109, A052485, A059404, A087207, A328400.

Programs

Mathematica

Array[Total@ Map[2^(# - 1) &, Union[FactorInteger[#][[All, -1]] ]] - Boole[# == 1] &, 86] (* Michael De Vlieger, Dec 29 2017 *)

PARI

a(n) = my (x=Set(factor(n)[,2]~)); sum(i=1, #x, 2^(x[i]))/2

Formula

a(p^k) = 2^(k-1) for any prime number p and k > 0.

a(n^2) = A000695(2 * a(n)) / 2 for any n > 0.

a(n) <= 1 iff n is squarefree (A005117).

a(n) <= 3 iff n is cubefree (A004709).

a(n) is odd iff n belongs to A052485 (weak numbers).

a(n) is even iff n belongs to A001694 (powerful numbers).

a(n) AND 2 = 2 iff n belongs to A038109 (where AND denotes the bitwise AND operator).

A000120(a(n)) <= 1 iff n belongs to A072774 (powers of squarefree numbers).

A000120(a(n)) > 1 iff n belongs to A059404.

If gcd(m, n) = 1, then a(m * n) = a(m) OR a(n) (where OR denotes the bitwise OR operator).

a(n) = a(A328400(n)). - Peter Munn, Oct 02 2023