Those Fascinating Numbers 107

834

• the rank of the prime number which appears the most often as the 15th prime

factor of an integer: p834 = 6397 (see the number 199).

836

• the second bizarre number (see the number 70);

• the fourth number which is not a palindrome, but whose square is a palindrome

(see the number 26).

839

• the smallest prime factor of the Mersenne number 2419 − 1.

840 (=

23

· 3 · 5 · 7)

• the largest number n such that

pn

p |n

1

p

1: here this sum is approximately

0.997975. . . ;

• the

15th

highly composite number (see the number 180).

841 (= 292)

• the smallest powerful number which can be written as the sum of two co-prime

3-powerful numbers = 1 (a number n is said to be k-powerful if p|n =⇒ pk|n):

here

292

= 841 = 216 + 625 =

23

·

33

+

54;

the sequence of numbers satisfying

this property begins as follows: 841, 968, 2312, 3528, 5041, 5776, 12769, 14884,

16641, 45125, 51984, . . .

843

• the smallest number n such that

f(n + 1) = f(n + 2) = f(n + 3) = f(n + 4),

where f(n) stands for the product of the exponents in the factorization of n:

here the common value of f(n + i) is 2; if for each number k ≥ 2, we denote by

nk the smallest positive integer n such that

f(n + 1) = f(n + 2) = . . . = f(n + k),

we then have the following table:

k 2 3 4 5 6 7

nk 1 4 843 74 848 671 345 8 870 024