# A000396 and related

Antreas P. Hatzipolakis xpolakis at otenet.gr
Sat Jun 3 16:33:06 CEST 2000

```ID Number: A000396 (Formerly M4186 and N1744)
Sequence:  6,28,496,8128,33550336,8589869056,137438691328,
2305843008139952128,2658455991569831744654692615953842176,
191561942608236107294793378084303638130997321548169216
Name:      Perfect numbers: equal to sum of proper divisors.
Formula:   The numbers 2^(p-1)(2^p - 1) are perfect, where p is a prime such
that 2^p- 1 is also prime (see A000043, the Mersenne primes), and
it is believed that there are no other perfect numbers.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
there are no ODD perfect numbers.

_________________________________________________________________________

Perfect            Number of Digits     Number of Aliquot Divisors
_________________________________________________________________________
6                           1                         3
28                          2                         5
496                         3                         9
8128                        4                        13
33550336                    8                        25
8589869056                 10                        33
137438691328               12                        37
2^30(2^31 - 1)             19                        61
2^60(2^61 - 1)             37                       121
2^88(2^89 - 1)             54                       177
2^106(2^107 - 1)           65                       213
2^126(2^127 - 1)           77                       253
2^520(2^521 - 1)          314                      1041
2^606(2^607 - 1)          366                      1213
2^1278(2^1279 - 1)        770                      2557
2^2202(2^2203 - 1)       1327                      4405
2^2280(2^2281 - 1)       1373                      4561
2^3216(2^3217 - 1)       1937                      6433
2^4252(2^4253 - 1)       2561                      8505
2^4422(2^4423 - 1)       2663                      8845
2^9688(2^9689 - 1)       5834                     19377
2^9940(2^9941 - 1)       5985                     19881
2^11212(2^11213 - 1)     6751                     22425
2^19936(2^19937 - 1)    12003                     39873
.........               .....                     .....

Reference:
New York: Dover Publications, Inc., 1979, p. 149
(1st publ. by Charles Scribner's Sons, New York, 1966, under the title:
Mathematics on Vacation)

Antreas

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