Those Fascinating Numbers 75

that | sin n1| | sin n2| . . . | sin nk| . . .; the sequence of numbers sa-

tisfying this property begins as follows: 1, 3, 22, 333, 355, 103 993, 104 348,

208 341, 312 689, 833 719, 1 146 408, 4 272 943, 5 419 351, 80 143 857, . . . (see also

the number 344)80.

336

• the smallest number m such that equation σ(x) = m has exactly eight solutions,

namely 132, 140, 182, 188, 195, 249, 287 and 299.

337

• the eighth prime number pk such that p1p2 . . . pk − 1 is prime (see the number

317).

338

• the smallest number n which can be written as the sum of the squares of two

prime numbers in two distinct ways: 338 =

72

+

172

=

132

+

132;

if nk stands

for the smallest number which can be written as the sum of the squares of two

prime numbers in k distinct ways, then n2 = 338, n3 = 2 210, n4 = 10 370,

n5 = n6 = n7 = n8 = 81 770 and n9 = n10 = n11 = n12 = n13 = 9 549 410;

• the smallest number n having at least two distinct prime factors and such that

p|n =⇒ p + 12|n + 12; the sequence of numbers satisfying this property begins

as follows: 338, 828, 1458, 4563, 4608, 5476, 6125, 6498, 7268, 7968,. . .

340

• possibly the largest number n such that n(n + 1)(n + 2)(n + 3) has exactly the

same prime factors as m(m + 1)(m + 2)(m + 3) for a certain number m n:

here m = 152 and the prime factors common to these two quantities are 2, 3,

5, 7, 11, 17, 19 and 31, since

152 · 153 · 154 · 155 =

24

·

32

· 5 · 7 · 11 · 17 · 19 · 31,

340 · 341 · 342 · 343 =

23

·

32

· 5 ·

73

· 11 · 17 · 19 · 31;

• the

13th

number n such that n! + 1 is prime (see the number 116).

80The

terms of the sequence (nk)k≥2 are simply the numerators of the convergents of the continued

fraction of π, which is given by

π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, . . .] = 3 +

1

7 +

1

15+

1

1+

1

292+...

,

the sequence of convergents being

3,

22

7

,

333

106

,

355

113

,

103993

33102

,

104348

33215

,

208341

66317

,

312689

99532

,

833719

265381

,

1146408

364913

,

4272943

1360120

,

5419351

1725033

,

80143857

25510582

, . . .