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].