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:
Joseph S. Madachy: Madachy's Mathematical Recreations.
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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