Those Fascinating Numbers 65

258

• the rank of the prime number which appears the most often as the

12th

prime

factor of an integer : p258 = 1 627 (see the number 199).

261

• the only three digit number n such that

2n

− n is prime (an observation due

to Meng Hsuan Wu (2002)): the only known numbers n for which the corre-

sponding number

2n

− n is prime are 13, 19, 21, 55, 261, 3 415, 4 185, 7 353 and

12 213.

263

• the second prime number p such that

79p−1

≡ 1 (mod

p2):

the only prime

numbers p

232

satisfying this congruence are 7, 263, 3 037, 1 012 573 and

60 312 841 (see Ribenboim [169], p. 347);

• the smallest prime factor of the Mersenne number

2131

− 1, whose complete

factorization is given by

2131

− 1 = 263 · 10350794431055162386718619237468234569.

264

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

positive powers of its digits: 264 = 21 + 61 + 44 = 25 + 63 + 42: the only other

number satisfying this property is 373;

• the largest three digit number = 100, 200, 300, and whose square contains no

more than two distinct digits: 2642 = 69 696 (see the number 109);

• the second number which is not a palindrome, but whose square is a palindrome

(see the number 26).

265

• the number of possible arrangements of the integers 1, 2, 3, 4, 5, 6 with the

restriction that the integer j must not be in the

jth

position for each j, 1 ≤

j ≤ 6: more generally, the number nk of possible arrangements of the integers

1, 2, . . . , k with the restriction that the integer j must not be in the

jth

position

for each j, 1 ≤ j ≤ k, is equal to

k!

1

2!

−

1

3!

+ . . . +

(−1)k

k!

,

and therefore n2 = 1, n3 = 2, n4 = 9, n5 = 44, n6 = 265, n7 = 1 854,

n8 = 14 833, n9 = 133 496, n10 = 1 334 961 and n11 = 14 684 570;