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A135656
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Perfect numbers divided by 2, written in base 2.
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1
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11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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).
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LINKS
| O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.
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FORMULA
| a(n)=A133028(n) written in base 2.
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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
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CROSSREFS
| Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.
Sequence in context: A110574 A191111 A127851 * A130602 A099814 A068053
Adjacent sequences: A135653 A135654 A135655 * A135657 A135658 A135659
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KEYWORD
| base,nonn
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AUTHOR
| Omar E. Pol (info(AT)polprimos.com), Feb 28 2008
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