76 Jean-Marie De Koninck
341
the smallest pseudoprime in base 2; the ten smallest pseudoprimes in base 2
are 341, 561, 645, 1 105, 1 387, 1 729, 1 905, 2 047, 2 465 and 2 701.
342
the smallest number n such that the Liouville function λ0 takes successively,
starting with n, the values 1, −1, 1, −1, 1, −1, 1, −1 (see the number 6 185).
344
the smallest solution of τ (n + 16) = τ (n) + 16; the sequence of numbers satis-
fying this equation begins as follows: 344, 488, 584, 704, 776, 794, 1034, 1064,
1208, 1334, . . . ;
the smallest number which can be written both as the sum of two cubes and
as the sum of three cubes: 344 =
13
+
73
=
43
+
43
+
63;
the sixth term of the sequence (nk)n≥1 defined as follows: n1 = 1 and for each
k 2, nk is the smallest number n such that | cos n| | cos nk−1|, so that
| cos n1| | cos n2| . . . | cos nk| . . .; the sequence of numbers satisfying
this property begins as follows: 1, 2, 5, 8, 11, 344, 699, 1 054, 1 409, . . . (see
also the number 333); it is interesting to
observe81
that nk = nk−1 + 355 for
each 7 k 152, while n153 = 260 515, n154 = 573 204, n155 = 4 846 147,
and then that nk = nk−1 + 5 419 351 for k = 156, 157, 158, 159, 160, 161, and
n162 = 122 925 461.
345
the smallest odd number n 1 such that σ3(n) is a perfect square: indeed,
σ3(345) = 65522; the numbers n 108 such that σ3(n) is a perfect square are
1, 2, 345, 690, 47 196, 46 248 900, 53 262 468 and 71 315 748.
347
the fourth self contained number (see the number 293);
the smallest prime number q such that
1
7
+
1
11
+
1
13
+ . . . +
1
q
1; if we denote
by qk the smallest prime number q such that
1
pk
+
1
pk−1
+ . . . +
1
q
1, then the
sequence (qk)k≥1 begins as follows: 5, 29, 109, 347, 857, 1627, 2999, 4931, 7759,
11677, . . . (see the number 857 for an estimate of the size of qk with respect
to k);
the smallest number n0 such that inequality π(kn) kπ(n) holds for all n n0,
for any real number k

e (see C. Karanikolov [115]).
81This
is certainly related to the fact that the fraction 355/113 is an excellent approximation of
π (in fact better than 22/7).
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