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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