Those Fascinating Numbers 173

• the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) = 11: here

6 578 = 2 · 11 · 13 · 23, 6 579 = 32 · 17 · 43 and 6 580 = 22 · 5 · 7 · 47 (see the number

2 210).

6 601 (= 7 · 23 · 41)

• the sixth Carmichael number (see the number 561).

6 603

• the only known number n 9 (besides 89) such that n = d1 +d2 2 +d3 4 +d4 8 +. . .+

drr−1

2

, where d1, d2, . . . , dr stand for the digits of n: here 6 603 =

6+62 +04 +38.

6 611

• the fourth number n 1 such that n · 2n + 1 is prime (see the number 141).

6 694

• the third number n such that

∑

p≤pn

p is a perfect square: here

∑

p≤p6694

p =

14 5732 (see the number 2 474).

6 788

• the smallest number of persistence 6 (see the number 679).

6 801

• the smallest number n such that the decimal expansion of 2n contains seven

consecutive zeros (see the number 53).

6 853

• the fifth number n such that n, n + 1, n + 2 and n + 3 have the same number

of divisors, namely eight (see the number 242).

6 859

• the smallest number n such that P (n)3|n and P (n + 1)3|(n + 1): here 6 859 =

193 and 6860 = 22 · 5 · 73; the sequence of numbers satisfying this proper-

ty begins as follows: 6859, 11859210, 18253460, 38331320423, 41807225999,

50788425848, . . . ; if nk stands for the smallest number n such that P (n)k|n

and P (n + 1)k|(n + 1), then n2 = 8, n3 = 6 859 and n4 = 11 859 210, while

n5 ≤ 437 489 361 912 143 559 513 287 483 711 091 603 378 (for three consecutive,

see the number 1 294 298).