Those Fascinating Numbers 145

2 311

• the fifth prime number p of the form p =

r

i=1

pi + 1: 2 311 = 2 · 3 · 5 · 7 · 11 + 1

(see the number 379).

2 312

• the third powerful number which can be written as the sum of two co-prime

3-powerful numbers = 1: 2 312 = 125 + 2 187, that is

23

·

172

=

53

+

37

(see the

number 841).

2 318

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 15 (see the

number 177).

2 319

• the fourth number n such that

2n

− 7 is prime (see the number 39).

2 329

• the smallest number n such that

φ11(n)

= 2, where

φ11(n)

stands for the

11th

iteration of the φ function (see the number 137).

2 351

• the smallest prime factor of the Mersenne number 247 − 1 (Euler, 1741), whose

complete factorization is given by

247

− 1 = 140 737 488 355 327 = 2351 · 4513 · 13264529;

• (probably) the largest number which cannot be written as the sum of two co-

prime numbers whose index of composition is ≥ 1.6 (see the number 933).

2 352

• the second solution of σ(n) = 3n + 12 (see the number 780).

2 377

• the 12th known prime number pk such that p1p2 . . . pk − 1 is prime (see the

number 317).