Those Fascinating Numbers 253

290 783

• the smallest number n such that P (n + i) ≤

√

n + i for i = 0, 1, 2, 3, 4, 5, 6; the

largest prime factors of these seven are respectively 271, 233, 311, 419,

227, 523 and 269, all smaller than

√numbers

290783 ≈ 539; it is also the smallest number

n such that P (n + i) ≤

√

n + i for i = 0, 1, 2, 3, 4, 5, 6, 7, since P (290790) = 359

(see the numbers 1 518 and 134 848).

293 760 (=

27

·

33

· 5 · 17)

• the smallest solution of

σ(n)

n

=

15

4

; the sequence of numbers satisfying this

equation begins as follows: 293 760, 1 782 144, 3 485 664, 282 977 280,

1 716 728 832,. . .

294 001

• the smallest prime number such that if one replaces any of its digits by any

another digit, one obtains a composite number.

297 864

• the smallest Niven number n such that n +60 is also a Niven number, but with

no others in between (see the number 28 680).

301 140

• the 11th number n such that σ(n) and σ2(n) have the same prime factors,

namely the primes 2, 3, 5, 7 and 13 (see the number 180).

310 154

• the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) = 14: here

310 154 = 2·13·79·151, 310 155 = 3·5·23·29·31 and 310 156 = 22 ·7·11·19·53

(see the number 2 210).

319 489

• the smallest prime factor of the Fermat number F11 =

2211

+1, whose complete

factorization is given by

F11 = 319489·974849·167988556341760475137·3560841906445833920513·P564.

322 033 (= 251 · 1283)

• the 100

000th

number having exactly two distinct prime factors (see the number

184).