A135650 Even perfect numbers written in base 2.

110, 11100, 111110000, 1111111000000, 1111111111111000000000000, 111111111111111110000000000000000, 1111111111111111111000000000000000000

Offset: 1

Author

Omar E. Pol, Feb 21 2008, Feb 22 2008, Apr 28 2009

Keywords

• base
• nonn

Comments

The number of digits of a(n) is equal to 2*A000043(n)-1. The central digit is "1". The first digits are "1". The last digits are "0". The number of digits "1" is equal A000043(n). The number of digits "0" is equal A000043(n)-1.

The concatenation of digits "1" of a(n) gives the n-th Mersenne prime written in binary (see A117293(n)).

Also, the number of digits of a(n) is equal to A133033(n), the number of proper divisors of n-th even perfect number.

Examples

			a(3) = 111110000 because the 3rd even perfect number is 496 and 496 written in base 2 is 111110000. Note that 11111 is the 3rd Mersenne prime A000668(3) = 31 written in base 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Crossrefs

Cf. A000043, A000396, A000668, A090748, A117293.

Cf. A061645, A133033.

Programs

Mathematica

Map[FromDigits[IntegerDigits[#, 2]] &, PerfectNumber[Range[8], "Even"]] (* Amiram Eldar, Oct 18 2024 *)