Those Fascinating Numbers 243

159 135

• the smallest number n such that

∑

m≤n

φ(m) is a multiple of 100 000; here the

sum is equal to 7 697 600 000.

159 841 (=

112

· 1321)

• the smallest 10-hyperperfect number: we say that a number n is 10-hyper-

perfect if it can be written as n = 1 + 10

d|n

1dn

d (which is equivalent to the

condition 10σ(n) = 11n + 9).

160 001

• the sixth prime number of the form n4 + 1 (see the number 1 297).

160 225 (=

52

· 13 · 17 · 29)

• the smallest number which can be written as the sum of two squares in 11

distinct ways (as well as in 12 distinct ways), namely 160 225 =

152

+

4002

=

322

+

3992

=

762

+

3932

=

812

+

3922

=

1132

+

3842

=

1402

+

3752

=

1752

+

3602

=

1832

+

3562

=

2162

+

3372

=

2282

+

3292

=

2522

+

3112

=

2652

+

3002

(see the number 50).

161 038 (= 2 · 73 · 1103)

• the smallest even number n such that

2n

≡ 2 (mod n) (discovered by D.N. Leh-

mer in 1949): N.G.W.H. Berger [20] proved that there exist infinitely many even

numbers n such that

2n

≡ 2 (mod n); the sequence of numbers satisfying this

property begins as follows: 161 038, 215 326, 2 568 226, 3 020 626, 7 866 046,

9 115 426, 49 699 666, 143 742 226, 161 292 286, 196 116 194, 209 665 666,

213 388 066, 293 974 066, 336 408 382, 377 994 926, 410 857 426, 665 387 746,

667 363 522, 672 655 726, 760 569 694, . . .

161 568

• the smallest number which can be written as the sum of three distinct cubes

in five distinct ways:

161 568 =

23

+

163

+

543

=

93

+

153

+

543

=

173

+

393

+

463

=

183

+

193

+

533

=

263

+

363

+

463

(see the number 1 009).

162 401 (= 17 · 41 · 233)

• the smallest pseudoprime in bases 2, 3, 5, 7, 11 and 13.