Sub-perfect Numbers

Discussion in 'Number Theory' started by Murray Cantor, Jan 15, 2024.

  1. Murray Cantor

    Murray Cantor

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    Let's call a number N 'sub-perfect' if the sum of its divisors less than N is N-1. (There may be a different term for such numbers). It is easy to see that powers of 2 are sub-perfect. Does anyone know a proof or counter-example to the conjecture that all sub-perfect numbers are powers of 2?
     
    Murray Cantor, Jan 15, 2024
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  2. Murray Cantor

    HallsofIvy

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    9=3*3 and 3+ 3= 6< 8= 9- 1.
     
    HallsofIvy, Jan 16, 2024
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  3. Murray Cantor

    Phrzby Phil

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    The divisors of 9 that are < 9 are 1 and 3. 1+3 = 4, which not= 8. So this is not a counter-example.
     
    Last edited: Jan 16, 2024
    Phrzby Phil, Jan 16, 2024
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  4. Murray Cantor

    Phrzby Phil

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    I think it can be proved by viewing numbers' binary representations.

    E.g., 7 = 111 (base 2) = 1+2+4, the divisors of 8 less than 8, which is 7+1.
     
    Phrzby Phil, Jan 16, 2024
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  5. Murray Cantor

    Phrzby Phil

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    I assumed that the original poster would comment on my approach.
     
    Phrzby Phil, Jan 23, 2024
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