Those Fascinating Numbers 247

203 433

• the smallest number n such that n, n + 1 and n + 2 are square-free and each

have four prime factors: 203 433 = 3 · 19 · 43 · 83, 203 434 = 2 · 7 · 11 · 1321,

203 435 = 5 · 23 · 29 · 61 and 203 436 = 5 · 23 · 29 · 61; the sequence of num-

bers satisfying this property begins as follows: 203433, 214489, 225069, 258013,

294593, 313053, 315721, 352885, 389389, 409353, . . . (see the number 1 309).

205 097

• the smallest number n requiring 12 iterations of the σI (n) function in order to

reach 1, which gives rise to the sequence 205098, 136736, 4274, 2138, 1070, 648,

121, 133, 160, 6, 4, 1 (see the number 193).

205 128

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

(see the number 102 564).

205 206

• the third solution of σ(n) = σ(n + 69) (see the number 8 786).

208 012

• the

12th

Catalan number (see the number 14).

208 335

• the largest number which is both triangular and square pyramidal: a number

is said to be square pyramidal if it can be written as the sum of the first m

squares for some number m: here

208 335 = 1 + 2 + 3 + . . . + 645 =

12

+

22

+

32

+ . . . +

852;

the only numbers which are both triangular and square pyramidal are 1, 55, 91

and 208 335 (for an important reference, see the number 645).

210 101

• the smallest six digit prime number whose digits are consecutive (see the num-

ber 67).

211 673

• the smallest number n such that Ω(n) = Ω(n + 1) = . . . = Ω(n + 6): here the

common value is 3 (see the number 602).