Those Fascinating Numbers 245

178 101

• the smallest number which is not a fifth power, but which can be written as the

sum of the fifth powers of some of its prime factors: here 178 101 =

32

· 7 · 11 ·

257 =

35

+

75

+

115

(see the number 870); at least five other numbers satisfy

this property, namely 3 077 237 625, 114 190 363 548 900, 1 067 104 211 610 690,

2 974 846 981 587 921 076 and 292 100 153 292 936 484 034.

179 487

• the fourth number which quadruples when its last digit is moved in first position

(see the number 102 564).

179 951

• the smallest prime factor of the Mersenne number

259

− 1, whose complete

factorization is given by

259

− 1 = 179951 · 3203431780337.

180 246

• the sixth number n such that 2n ≡ −2 (mod n) (see the number 946).

182 182

• the seventh number n divisible by a square 1 and such that δ(n+1)−δ(n) = 1

(see the number 49).

183 091

• the first term of the smallest sequence of ten consecutive prime numbers all of

the form 4n + 3 (see the number 463).

185 527

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

distinct prime numbers in two distinct ways: here

185 527 =

193

+

313

+

533

=

133

+

433

+

473;

if nk stands for the smallest number which can be written as the sum of the

cubes of three distinct prime numbers in k distinct ways, then n2 = 185 527,

n3 = 8 627 527 and n4 = 999 979 163.