Those Fascinating Numbers 79
368
the second solution of σ(n) = 2n + 8; the sequence of numbers satisfying
this equation begins as follows: 56, 368, 836, 11 096, 17 816, 45 356, 77 744,
91 388, 128 768, 254 012, 388 076, 2 087 936, 2 291 936, 13 174 976, 29 465 852,
35 021 696,. . .
83
369
the tenth number k such that k|(10k+1 1) (see the number 303).
370
one of the five numbers which can be written as the sum of the cubes of its
digits: 370 = 33 + 73 + 03; the others are 1, 153, 371 and 407.
371
one of the five numbers which can be written as the sum of the cubes of its
digits: 371 =
33
+
73
+
13;
the others are 1, 153, 370 and 407.
373
the largest three digit number (see the number 264) which can be written in two
distinct ways as the sum of positive powers of its digits: 373 =
31
+
73
+
33
=
34
+
72
+
35.
376
the smallest three digit automorphic number: 3762 = 141 376 (see the number
76).
377
the largest known Fibonacci pseudoprime; the only other one known is 323;
the smallest number n such that the decimal expansion of 2n contains four
consecutive zeros (see the number 53).
83It
is easy to show that any number n =

·p, where α is a positive integer such that p =
2α+1
−9
is prime, is a solution of σ(n) = 2n + 8: this is the case when α = 3, 4, 8, 10, 16, 20, 32 (and for no
other values of α 100); the solutions corresponding to α = 3, 4, 8, 10 are included in the above
list. It is also possible to identify the solutions n of the form · p · q, with p q primes and α
a positive integer. This is how one obtains the solutions n = 836, 11 096, 17 816, 77 744, 2 291 936,
13 174 976 and 35 021 696 listed here.
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