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).