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
, . . .
Previous Page Next Page