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A perfect number equals the sum of its proper divisors (excluding itself). The first perfect numbers are 6, 28, 496, 8128. All known perfect numbers are even; it is unknown whether odd perfect numbers exist.
Formula
n is perfect if σ(n) = 2n, where σ(n) is sum of all divisors
- n
- the number being tested
- σ(n)
- sum of all divisors of n (including n) (integer)
Guida passo passo
- 1Sum all divisors except n itself
- 26: 1+2+3 = 6 ✓
- 328: 1+2+4+7+14 = 28 ✓
- 4Abundant: divisor sum > n; Deficient: divisor sum < n
Esempi risolti
Ingresso
n = 28
Risultato
Divisors 1,2,4,7,14 — Sum = 28 (Perfect!)
Ingresso
n = 12
Risultato
Divisors 1,2,3,4,6 — Sum = 16 > 12 (Abundant)
Domande frequenti
What is the smallest perfect number?
6: its divisors are 1, 2, 3, 6, and 1+2+3 = 6 (sum of proper divisors equals n).
Is 28 a perfect number?
Yes: divisors are 1, 2, 4, 7, 14, 28, and 1+2+4+7+14 = 28.
How many perfect numbers are known?
51 perfect numbers are known (as of 2024). All known ones are even and quite rare.
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