Those Fascinating Numbers 157

3 605

• the number of Carmichael numbers 1011 (see the number 646).

3 610

• the

20th

number n such that n! − 1 is prime (see the number 166).

3 647

• the smallest number n for which the Moebius function µ takes successively,

starting with n, the values 1,0,1,0,1,0,1,0; if we denote by nk the smallest

number n for which the Moebius function µ takes successively, starting with

n, the values 1, 0, 1, 0, . . . , 1

or

0,

k

numbers

then we have n2 = 15, n3 = 55, n4 = 159,

n5 = n6 = n7 = 411, n8 = 3 647, n9 = n10 = n11 = 15 243, n12 = 113 343,

n13 = n14 = n15 = 1 133 759 and n16 = 29 149 139: one can easily prove that

there are no146 numbers nk with k ≥ 17 (for the sequence −1, 0, −1, 0, . . ., see

the number 2 749).

3 655

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

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

3 675 (= 3 ·

52

·

72)

• the smallest odd number n having exactly three prime factors and such that

γ(n)|φ(n).

3 684

• the 13th Keith number (see the number 197).

3 705

• the

15th

solution of φ(n) = φ(n + 1) (see the number 15).

3 759

• the seventh number such that

2n

+

n2

is prime (see the number 2 007).

146This simply follows from the fact that any sequence of 17 consecutive numbers, the first of

which is square-free, must include two numbers r and s both divisible by 9 and therefore such that

µ(r) = µ(s) = 0 with r − s odd.