126 Jean-Marie De Koninck

1 352

• the only solution n

1012

of σ(n) = 2n + 41 (see the number 196).

1 361

• the smallest prime number that is preceded by 33 consecutive composite num-

bers; indeed, there are no prime numbers between 1 327 and 1 361;

• the third prime number obtained by the Mills formula qk =

[θ3k

]: indeed,

Mills [136] proved that there exists a real number θ such that

[θ3k

] is a prime

number for k = 1, 2, 3, . . ., this constant being θ = 1.30637788386308069 . . .:

the first four terms of the sequence of numbers (qk)k≥1 are: 2, 11, 1 361 and

2 521 008 887.

1 364

• the fifth solution of σ(n) = σ(n + 1) (see the number 14).

1 367

• the smallest prime factor of the Mersenne number

2683

− 1;

• the eighth prime number p such that

(3p

− 1)/2 is itself a prime number (see

the number 1 091).

1 368

• the number of ways one can fold a 3 × 3 stamp sheet: for n = 1, 2, 3, 4, 5,

the number of ways one can fold an n × n stamp sheet is 1, 8, 1 368, 300 608,

186 086 600, respectively (see Sloane & Plouffe [188], sequence M4587).

1 375

• the smallest number which is the first of three consecutive numbers each being

divisible by a cube 1: 1 375 =

53

· 11, 1 376 =

25

· 43, 1 377 =

34

· 17; if nk

stands for the smallest number which is the first of three consecutive numbers

each being divisible by a

kth

power, then n2 = 48, n3 = 1 375, n4 = 33 614,

n5 = 2 590 623, n6 = 26 890 623 and n7 = 2 372 890 624;

• the largest known number n such that γ(n + 1) − γ(n) = 31; the only solutions

n 109 of this equation are 32, 36, 40, 45, 60 and 1 375 (see the number 98).

1 385

• the eighth Euler number (see the number 272).