Those Fascinating Numbers 115

977

• the fourth Stern number (see the number 137).

983

• the smallest prime factor of the Mersenne number 2491 − 1, whose complete

factorization is given by

2491

− 1 = 983 · 7707719 · 110097436327057 · 6976447052525718623

·19970905118623195851890562673

·3717542676439779473786876643915388439

·14797326616665978116353515926860025681383;

it is the smallest Mersenne number with exactly seven prime factors (see the

number 223 for the list of the smallest Mersenne numbers which require a given

number of prime factors).

985

• the 13th Markoff number (see the number 433).

987

• the smallest number n such that ω(n)+ω(n+1)+ω(n+2)+ω(n+3) = 12: here

987 = 3·7·47, 988 = 22 ·13·19, 989 = 23·43 and 990 = 2·32 ·5·11; if we denote

by nk the smallest number n such that ω(n)+ω(n+1)+ω(n+2)+ω(n+3) = k,

we have the following table:

k nk

4 2

5 3

6 9

7 12

8 33

k nk

9 75

10 153

11 492

12 987

13 4 179

k nk

14 13 803

15 18 444

16 134 043

17 282 489

18 1 013 724

k nk

19 4 289 592

20 12 582 633

21 57 495 513

22 260 628 717

23 801 621 093

991

• the ninth prime number pk such that p1p2 . . . pk − 1 is prime (see the number

317);

• the largest three digit number for which each permutation of its digits provides

a prime number, namely in this case the prime numbers 991, 199 and 919; the

only other known prime numbers satisfying this property are 2, 3, 5, 7, 11, 13,

17, 31, 37, 71, 73, 79, 97 as well as all the primes of the form 11 . . . 1

k

(for these

last primes, see the number 19).