90 Jean-Marie De Koninck

491

• the smallest prime number p such that the number of irregular pairs (p, 2k)

is equal to 3: given an irregular prime p, we say that the pair (p, 2k) is an

irregular pair if 2 ≤ 2k ≤ p − 3 and if p divides the numerator of the Bernoulli

number B2k (see Ribenboim [169], p. 347);

• the smallest number n such that P (n) P (n + 1) . . . P (n + 4): here

491 41 29 19 11 (see the number 1 851).

492

• the largest number n such that n

pn

log pn − 1

:

here 492

3527

log 3527 − 1

≈ 492.03 (see the number 61).

495

• the value of the Kaprekar constant for the three digit numbers, namely the

number that one obtains by the Kaprekar process: given a three digit number n

which is not a palindrome, consider the numbers a and b obtained respectively

by placing the digits of n in decreasing order and in increasing order, then

consider the number a − b, and repeat the process, until the number 495 is

reached, thus completing the Kaprekar algorithm; more generally, given an

arbitrary integer k ≥ 2, applying the above algorithm to any k digit number

will eventually yield (repetitively) some number ck, which is called the Kaprekar

constant for the k digit numbers; the sequence (ck)k≥2 begins as follows: 63,

495, 6174, 99954, . . . (D.R. Kaprekar [114]);

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 12 (see the

number 177);

• the eighth solution of φ(n) = φ(n + 1) (see the number 15).

496

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

• the ninth Granville number (see the number 126).

503

• the smallest prime factor of the Mersenne number

2251

− 1, whose complete

factorization is given by

2251

− 1 = 503 · 54217 · 178230287214063289511

·61676882198695257501367 · 12070396178249893039969681.