Number_Theory

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Perfect Numbers

The following procedure will search for perfect numbers less than 10⁶:

for n in range(1,10^6):
    if sigma(n)==2*n:
        print(n)

Mersenne primes and perfect numbers

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The following procedure searches for Mersenne primes among the first 20 prime exponents.  The output includes the exponent, the Mersenne prime, and the corresponding perfect number.

The following procedure searchs for Mersenne primes among the first 20 prime exponents. The output includes the exponent, the Mersenne prime, and the corresponding perfect number.

P=Primes()
for n in range(0,19):
    p=P.unrank(n)
    if is_prime(2^p-1):
        print(p,2^p-1,2^(p-1)*(2^p-1))

(2, 3, 6)
(3, 7, 28)
(5, 31, 496)
(7, 127, 8128)
(13, 8191, 33550336)
(17, 131071, 8589869056)
(19, 524287, 137438691328)
(31, 2147483647, 2305843008139952128)
(61, 2305843009213693951, 2658455991569831744654692615953842176)

Product Expansion for Proposition 15.51

f=expand(prod(1-x^k for k in range(1,51)))
table([(j,(f.coefficient(x,j))) for j in [1..51]], header_row=["n", "Coeff. of x^n"], frame=True)

+----+---------------+
| n  | Coeff. of x^n |
+====+===============+
| 1  | -1            |
+----+---------------+
| 2  | -1            |
+----+---------------+
| 3  | 0             |
+----+---------------+
| 4  | 0             |
+----+---------------+
| 5  | 1             |
+----+---------------+
| 6  | 0             |
+----+---------------+
| 7  | 1             |
+----+---------------+
| 8  | 0             |
+----+---------------+
| 9  | 0             |
+----+---------------+
| 10 | 0             |
+----+---------------+
| 11 | 0             |
+----+---------------+
| 12 | -1            |
+----+---------------+
| 13 | 0             |
+----+---------------+
| 14 | 0             |
+----+---------------+
| 15 | -1            |
+----+---------------+
| 16 | 0             |
+----+---------------+
| 17 | 0             |
+----+---------------+
| 18 | 0             |
+----+---------------+
| 19 | 0             |
+----+---------------+
| 20 | 0             |
+----+---------------+
| 21 | 0             |
+----+---------------+
| 22 | 1             |
+----+---------------+
| 23 | 0             |
+----+---------------+
| 24 | 0             |
+----+---------------+
| 25 | 0             |
+----+---------------+
| 26 | 1             |
+----+---------------+
| 27 | 0             |
+----+---------------+
| 28 | 0             |
+----+---------------+
| 29 | 0             |
+----+---------------+
| 30 | 0             |
+----+---------------+
| 31 | 0             |
+----+---------------+
| 32 | 0             |
+----+---------------+
| 33 | 0             |
+----+---------------+
| 34 | 0             |
+----+---------------+
| 35 | -1            |
+----+---------------+
| 36 | 0             |
+----+---------------+
| 37 | 0             |
+----+---------------+
| 38 | 0             |
+----+---------------+
| 39 | 0             |
+----+---------------+
| 40 | -1            |
+----+---------------+
| 41 | 0             |
+----+---------------+
| 42 | 0             |
+----+---------------+
| 43 | 0             |
+----+---------------+
| 44 | 0             |
+----+---------------+
| 45 | 0             |
+----+---------------+
| 46 | 0             |
+----+---------------+
| 47 | 0             |
+----+---------------+
| 48 | 0             |
+----+---------------+
| 49 | 0             |
+----+---------------+
| 50 | 0             |
+----+---------------+
| 51 | 2             |
+----+---------------+


︠5e4525e3-b0c9-4ea2-867d-04159e7a25cd︠

Chapter 15 - Arithmetic Functions

Perfect Numbers

Mersenne primes and perfect numbers

Product Expansion for Proposition 15.51

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