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).
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