A135656 - OEIS

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A135656

Perfect numbers divided by 2, written in base 2.

11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000

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OFFSET

1,1

COMMENTS

The number of divisors of a(n) is equal to the number of its digits. This number is equal to 2*A000043(n)-2. The number of divisors of a(n) that are powers of 2 is equal to the number of divisors that are multiples of n-th Mersenne prime A000668(n) and this number of divisors is equal to A090748(n). The first digits of a(n) are "1". For n>1 the last digits are "0". The number of digits "1" is equal to A000043(n). The number of digits "0" is equal to A000043(n)-2. The concatenation of digits "1" gives the n-th Mersenne prime written in binary (see A117293(n)). The structure of divisors of a(n) represent a triangle (see example).

LINKS

Table of n, a(n) for n=1..8.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

FORMULA

a(n)=A133028(n) written in base 2.

EXAMPLE

a(4)=111111100000 because the 4th. perfect number is 8128 and 8128/2=4064 and 4064 written in base 2 is 111111100000. Note that 1111111 is the 4th. Mersenne prime A000668(4)=127, written in base 2.

The structure of divisors of a(4)=111111100000

CROSSREFS

Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.

Sequence in context: A287044 A191111 A127851 * A130602 A362919 A289372

Adjacent sequences: A135653 A135654 A135655 * A135657 A135658 A135659

KEYWORD

base,nonn,less

AUTHOR

Omar E. Pol, Feb 28 2008

STATUS

approved