48 Jean-Marie De Koninck
157
the largest prime number p 10 000 such that
(pp
+ 1)/(p + 1) is a prime
number; the only known prime numbers satisfying this property are 3, 5, 17
and 157;
the second prime number equally distant, by a distance of 6, from the preceding
and following prime numbers: p36 = 151, p37 = 157 and p38 = 163 (the smallest
prime number satisfying this property is 53);
the smallest odd number which is not the sum of four non zero distinct squares
(F. Halter-Koch; see R K. Guy [101], C20);
the smallest number n such that φ(2n + 1) φ(2n): the sequence of numbers
satisfying this inequality begins as follows: 157, 262, 367, 412, 472, 487, 577,
682, 787, 877, 892, 907, 997, 1207, 1237, 1567, 1627, 1657,. . .
158
the second solution of σ(n) = σ(n + 19) (see the number 34).
159
the smallest number n such that the Moebius function µ takes successively,
starting with n, the values 1, 0, 1, 0: the sequence of numbers satisfying this
property begins as follows: 159, 247, 303, 339, 411, 413, 685, 721, 849, 949,
. . . (see the number 3 647).
161
the second solution of σ(n) = σ(n + 30) (see the number 88).
162
the only number n
1012
such that σ(n) = 2n + 39 (see the number 196).
163
the largest Heegner number: a number d is called a Heegner number if d 0 is
square-free and if the factorization in the set Ed = {a + b

−d : 2a, 2b Z} is
unique up to a unit (in the cases d = 1 and d = 2, the set Ed is instead defined
by Ed = {a + b

−d : a, b Z}); there exist exactly nine Heegner numbers,
namely 1, 2, 3, 7, 11, 19, 43, 67 and 163 (see J.H. Conway & R.K. Guy [36]);
the only solution “SIX” of the French cryptarithm
DIX2

SIX2
= SEIZE × 4,
2632

1632
= 10650 × 4
(see Michel Criton [39]);
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