18 Jean-Marie De Koninck

50

• the smallest number which can be written as the sum of two squares in two

distinct ways: 50 = 12 +72 = 52 +52; if nk stands for the smallest number which

can be written as the sum of two squares in k distinct ways, then n2 = 50,

n3 = 325, n4 = 1 105, n5 = n6 = 5 525, n7 = n8 = 27 625, n9 = 71 825,

n10 = 138 125 and n11 = n12 = 160 225 (see also the number 1

729)25;

• the smallest number n having at least two distinct prime factors and which

is such that p|n =⇒ p + 10|n + 10; the sequence of numbers satisfying this

property begins as follows: 50, 242, 245, 935, 1250, 8405, . . .

51

• the smallest solution of σ(n) = σ(n + 4); the sequence of numbers satisfying

this equation begins as follows: 51, 66, 115, 220, 319, 1003, 2585, 4024, 4183,

4195, 5720, 5826, 5959, 8004, 8374, . . . ;

• the fourth number n such that φ(n)σ(n) is a perfect square: here φ(51)σ(51) =

482;

the sequence of numbers satisfying this property begins as follows: 1, 14,

30, 51, 105, 170, 194, 248, 264, 364, 405, 418, 477, 595, 679, . . . ; Niegel Boston

proved that this sequence is infinite (see R.K. Guy & R.J. Nowakowski [103]);

• the second number n such that σ(n) = σ(n + 20) (see the number 42).

52

• the fifth Bell number, namely B5: Bell numbers Bn are defined implicitly by

eex

−1

=

∞

n=0

Bn

xn

n!

;

the first Bell numbers are B0 = 1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52,

B6 = 203, B7 = 877, B8 = 4 140, B9 = 21 147, B10 = 115 975;

• the smallest number k 5 such that equation σ(n) − n = k has no solution26;

• the largest known number27 n such that n! + n + 1 is prime; the others are 2,

4, 6 are 10.

53

• the smallest prime number equally distant, by a distance of 6, from the preced-

ing and following prime numbers : p15 = 47, p16 = 53 and p17 = 59;

• the smallest number n such that the decimal expansion of 2n contains two

consecutive zeros (see the sequence M4710 in N.J.A. Sloane & S. Plouffe [188]);

25It is natural to raise the following question: Does there exist a number k0 such that nk nk+1

for all k k0 ?

26Interestingly, Erd˝ os pointed out the existence of a large family of integers k for which equation

σ(n) − n = k has no solution.

27Obviously, such a number n must be such that n + 1 is prime.