Those Fascinating Numbers 271

823 544 (= 23 · 113 · 911)

• the seventh number of the form

nn

+ 1 (see the number 3 126).

825 265 (= 5 · 7 · 17 · 19 · 73)

• the smallest Carmichael number which is the product of five prime numbers:

it is the

40th

Carmichael number (see the number 41 041).

838 561

• the smallest prime number p such that Ω(p+1) = 2, Ω(p+2) = 3, Ω(p+3) = 4,

Ω(p + 4) = 5 and Ω(p + 5) = 6 (see the number 61).

846 137

• the smallest prime number amongst those which appear more often as the fourth

prime factor of an integer than as the third prime factor; more precisely, when

846 137 appears in the prime factorization of a number, then in 24.547684% of

the cases, it is as the fourth prime factor of that number, while it is as the third

one in 24.547676% of the cases, the fifth in 17.16% of the cases and the sixth

in only 9% of the

cases186.

859 433

• the exponent of the

33rd

Mersenne prime

2859 433

− 1 (Slowinski, 1993).

873 612

• the value of

11

+

22

+ . . . +

77

(see the number 3 413).

874 660 (=

22

· 5 · 101 · 433)

• the smallest number n which allows the sum

m≤n

Ω(m)=5

1

m

to exceed 1; if nk stands

for the smallest number n = nk which allows the sum

m≤n

Ω(m)=k

1

m

to exceed

1, then n1 = 5, n2 = 35, n3 = 402, n4 = 10 305, n5 = 874 660 and n6 =

395 096 427.

186This

characterization of 846 137 can be obtained from the results found in J.M. De Koninck &

G. Tenenbaum [63].