Those Fascinating Numbers 67

272

• the seventh Euler number: the sequence (En)n≥0 of Euler numbers is defined

as follows: En is the number of permutations of the integers 1, 2, . . . , n whose

digits are increasing, then decreasing and again increasing; thus, for n = 4, the

permutations of 1,2,3,4 which satisfy this property are 1324, 1423, 2314, 2413

and 3412, and this is why E4 = 5; setting E0 = 1, the first terms of the sequence

are E1 = 1, E2 = 1, E3 = 2, E4 = 5, E5 = 16, E6 = 61, E7 = 272, E8 = 1 385,

E9 = 7 936, E10 = 50 521, E11 = 353 792, E12 = 2 702 765, E13 = 22 368 256,

. . . ; D. Andr´ e proved in 1879 that the sequence of Euler numbers (En)n≥0 can

be defined implicitly by the formula tan

x

2

+

π

4

=

∞

n=0

En

xn

n!

.

273 (= 3 · 7 · 13)

• the smallest square-free composite number n such that p|n =⇒ p + 7|n + 7 (see

the number 399).

276

• the number appearing in the equation

672n

+

159n

+

834n

=

276n

+

951n

+

438n,

which holds when n = 1 or n = 2, even when one eliminates the first digit in

each of these terms, and even when one eliminates the second one, and finally

even when one eliminates the third digit; the same is true for each of the three

equations

618n

+

753n

+

294n

=

816n

+

357n

+

492n,

654n

+

132n

+

879n

=

456n

+

231n

+

978n,

852n

+

174n

+

639n

=

258n

+

471n

+

936n;

these equalities go back to 3 000 years and were known to the Chinese (see M.

Criton [39], p. 20).

277

• the smallest prime number q which allows the sum

p≤q

1

p

to exceed 2; if we

denote by qk the smallest prime number such that

p≤qk

1

p

k, then q1 = p3 = 5,

q2 = p59 = 277 and q3 = p361139 = 5 195 977; one can

prove74

that q4 ≈

1018

(see the number 1 307 for an analogue problem).

74Indeed,

using inequalities

log log q4 + B −

1

2

log2

q4

p≤q4

1

p

log log q4 + B +

1

2

log2

q4

,