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
,
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