Those Fascinating Numbers 177

7 912

• the sixth bizarre number (see the number 70).

7 919

• the 1

000th

prime number (see the number 541).

7 936

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

8 042

• the largest known number which cannot be written as a sum of less than seven

cubes (of non negative integers): here 8 042 =

53

+

63

+

63

+

83

+

93

+

113

+

173

(see E.W. Weisstein [201], p. 1918).

8 064 (= 27 · 32 · 7)

• the fifth number n having at least two distinct prime factors and such that

B1(n) = β(n)2: here 27 + 32 + 7 = (2 + 3 + 7)2 (see the number 144).

8 128

• the fourth perfect number (see the number 6);

• the

13th

Granville number (see the number 126).

8 169

• the number of twin prime pairs 106 (see the number 1 224).

8 190

• the seventh number which is not perfect or multi-perfect but whose harmonic

mean is an integer (see the number 140);

• the third Erd˝ os-Nicolas number (see the number 2 016).

8 191

• the fifth Mersenne prime: 8 191 =

213

− 1;

• the smallest counter example Mq =

2q

−1 to the statement “If Mq is a Mersenne

prime, then MMq is prime”; indeed, M8

191

is composite since it is divisible by

338 193 759 479;

• the largest known prime number p such that P

(p2

−1) ≤ 13 and P

(p2

−1) = 13

(see the number 4 801).