# Sub-perfect Numbers

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

1. ### 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
2. ### HallsofIvy

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9=3*3 and 3+ 3= 6< 8= 9- 1.

HallsofIvy, Jan 16, 2024
3. ### 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
4. ### 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
5. ### Phrzby Phil

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I assumed that the original poster would comment on my approach.

Phrzby Phil, Jan 23, 2024
6. ### RobertSmart

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The conjecture that all sub-perfect numbers are powers of 2 is indeed false. A counter-example to this conjecture is 12.

To show that 12 is a sub-perfect number, let's find the sum of its divisors less than 12:

1 + 2 + 3 + 4 + 6 = 16

As the sum of its divisors less than 12 is 16, which is equal to 12 - 1, 12 qualifies as a sub-perfect number.

However, 12 is not a power of 2. Thus, 12 serves as a counter-example to the conjecture.

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RobertSmart, May 8, 2024
7. ### Phrzby Phil

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16 does not equal to 12 - 1.

Phrzby Phil, May 8, 2024

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