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.
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