Those Fascinating Numbers 151

2 730

• the denominator of the Bernoulli number B12 = −

691

2730

.

2 737

• the smallest number n such that ι(n) = ι(n + 1) = . . . = ι(n + 6) = 1; here ι(n)

stands for the index of isolation of n and is defined by ι(n) = min

1≤m=n

P (m)≤P (n)

|n − m|,

that is the distance to the nearest integer whose largest prime factor is no

larger than that of n; if nk stands for the smallest number n such that ι(n) =

ι(n + 1) = . . . = ι(n + k − 1) = 1, then we have the following table:

k 6 7 8 9 10 11

nk 169 2 737 26 536 67 311 535 591 3 021 151

k 12 13 14 15

nk 26 817 437 74 877 777 657 240 658 785 211 337

2 749

• 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 nk stands for 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 n2 = n3 = 3, n4 = 17, n5 = n6 = 195,

n7 = 1 547, n8 = 2 749, n9 = n10 = n11 = 4 011, n12 = 462 649, n13 = n14 =

n15 = 580 547 and n16 = 74 406 035: one can easily prove that there are

no142

numbers nk with k ≥ 17 (for the sequence 1, 0, 1, 0, . . ., see the number 3 647).

2 821 (= 7 · 13 · 31)

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

2 834

• the

13th

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

2 835

• the fourth odd abundant number (see the number 945).

142This easily follows from the fact that in any sequence of 17 consecutive numbers, the first of

which is square-free, there must be two numbers r and s both divisible by 9 and thus such that

µ(r) = µ(s) = 0 with r − s odd.