Those Fascinating Numbers 109
863
the smallest prime factor of the Mersenne number 2431 1, whose complete
factorization is given by
2431
1 = 863 · 3449 · 36238481 · 76859369 · 558062249
·4642152737 · 142850312799017452169 · P70;
it is the smallest Mersenne number with exactly eight prime factors (see the
number 223 for the list of the smallest Mersenne numbers which require a given
number of prime factors);
the second prime number p such that 13p−1 1 (mod p2): the only prime
numbers p
232
satisfying this congruence are 2, 863 and 1 747 591 (see Riben-
boim [169], p. 347).
864
the smallest solution of σ2(n) = σ2(n + 12); the sequence of numbers satis-
fying this property begins as follows: 864, 1 290, 6 474, 5 685 114, 8 173 434,
13 381 626, . . . 115;
the second number n 2 such that
σ(n) + φ(n)
γ(n)2
is an integer (see the number
588).
870
the smallest number which is not the square of a prime number, but which
can be written as the sum of the squares of some of its prime factors116: here
870 = 2 · 3 · 5 · 29 = 22 + 52 + 292; the only numbers smaller than 1010 satisfying
this property are 870, 188 355, 298 995 972, 1 152 597 606 and 1 879 906 755; the
number 5 209 105 541 772 also satisfies this property (see also the number 378);
if nk, for k 2, stands for the smallest number which is not a kth power, but
which can be written as the sum of the kth powers of some of its prime factors,
then
n2 = 870 = 2 · 3 · 5 · 29 =
22
+
52
+
292,
n3 = 378 = 2 ·
33
· 7 =
23
+
33
+
73,
n4 = 107 827 277 891 825 604 =
22
· 3 · 7 · 31 · 67 · 18121 · 34105993
=
34
+
314
+
674
+
181214,
n5 = 178 101 =
32
· 7 · 11 · 257 =
35
+
75
+
115,
115It is easy to generate these solutions using those of equation σ2(n) = σ2(n + 2) (see the number
1 089) namely by examining the numbers n = 6m with (m, 6) = 1, for which we have σ2(6m) =
σ2(6m + 12), an equation which is equivalent to σ2(m) = σ2(m + 2).
116No
one has yet been able to prove or disprove that such a number n (that is with ω(n) 2
and such that n =

p|n
p2) exists. For more on this matter, see the results of J.M. De Koninck &
F. Luca [54].
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