4 Jean-Marie De Koninck 11 • the smallest prime number p such that 3p−1 ≡ 1 (mod p2): the only other prime number p 232 satisfying this congruence is p = 1 006 003 (see Riben- boim [169], p. 347)3 • the smallest number n which allows the sum i≤n 1 i to exceed 3 (see the number 83). 12 • the smallest pseudo-perfect number: we say that a number is pseudo-perfect if it can be written as the sum of some of its proper divisors: here 12 = 6 + 4 + 2 in 1976, Erd˝ os proved that the set of pseudo-perfect numbers is of positive density (see R.K. Guy [101], B2) • the smallest number m for which equation σ(x) = m has exactly two solutions, namely 6 and 11 • the only number n 1 such that σ(γ(n)) = n • the smallest sublime number: we say that a number n is sublime if τ (n) and σ(n) are both perfect numbers: here τ (12) = 6 and σ(12) = 28 this concept was introduced by Kevin Ford the only other known sublime number is 2126(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1). 13 • the second Wilson prime (see the number 5) • the prime number which appears the most often as the fourth prime factor of an integer, namely 31 times in 5005 (see the number 199) • the smallest prime number p such that 23p−1 ≡ 1 (mod p2): the only prime numbers p 232 satisfying this congruence are 13, 2 481 757 and 13 703 077 (see Ribenboim [169], p. 347) • the third horse number: we say that n is a horse number if it represents the number of possible results accounting for ties, in a race in which k horses participate thus, if Hk is the kth horse number, one can prove4 that Hk = k i=1 ik ⎛ ⎝ k−i j=0 (−1)j j + i j ⎞ ⎠ the first 20 terms of the sequence (Hk)k≥1 are 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632573, 28091567595, 526858348381, 10641342970443, 230283190977853, 5315654681981355, 130370767029135901, 3385534663256845323, 92801587319328411133 and 2677687796244384203115. 3As is the case for the Wieferich primes (see the number 1 093), it is not known if this sequence of numbers is infinite. 4A formula established by Charles Cassidy (Universit´ e Laval).

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