Those Fascinating Numbers 171

6 174 (= 2 · 32 · 73)

• the value of the Kaprekar constant for the four digit numbers (see the number

495);

• the fifth solution of φ(n) =

γ(n)2

(see the number 108).

6 185

• the smallest number n such that the Liouville function λ0 takes successively,

starting with n, the values 1, −1, 1, −1, 1, −1, 1, −1, 1, −1, 1, −1; if nk stands for

the smallest number n such that λ0(n + i) =

(−1)i,

for i = 0, 1, 2, . . . , k − 1,

then n2 = 4, n3 = 22, n4 = 49, n5 = n6 = n7 = 58, n8 = n9 = 342,

n10 = 1 230, n11 = n12 = 6 185, n13 = 4 9784, n14 = 79 800, n15 = n16 = n17 =

n18 = n19 = n20 = n21 = n22 = 99 826, n23 = 7 815 614, n24 = 11 435 684,

n25 = 19 370 102, n26 = 39 623 112, n27 = 46 025 769, n28 = n29 = 544 865 099

and n30 = 1 075 790 572.

6 200

• the sixth number which is not perfect or multi-perfect but whose harmonic

mean is an integer (see the number 140).

6 247

• the largest known number n such that

n!+2n

−1 is prime; the others are n = 1,

2, 3, 5, 7, 11, 167 and 2 609.

6 318

• the first of the seven smallest consecutive numbers at which the Ω(n) function

takes distinct values, namely here the values 7, 2, 6, 4, 3, 1 and 5 (see the

number 726).

6 361

• the smallest prime factor of the Mersenne number 253 − 1, whose complete

factorization is given by

253

− 1 = 6 361 · 69 431 · 20 394 401.

6 380

• the

18th

number n such that n! + 1 is prime (see the number 116); note that

6 380! + 1 is a 21 507 digit number.