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Perfect numbers divided by 2, written in base 2.
1

%I #7 Mar 11 2014 01:34:11

%S 11,1110,11111000,111111100000,111111111111100000000000,

%T 11111111111111111000000000000000,

%U 111111111111111111100000000000000000,111111111111111111111111111111100000000000000000000000000000

%N Perfect numbers divided by 2, written in base 2.

%C 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).

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

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

%e 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.

%e The structure of divisors of a(4)=111111100000

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

%K base,nonn,less

%O 1,1

%A _Omar E. Pol_, Feb 28 2008