Those Fascinating Numbers 193

15 243

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

starting with n, the values 1,0,1,0,1,0,1,0,1; it is also the smallest number n for

which the µ function takes successively the values 1,0,1,0,1,0,1,0,1,0 as well as

1,0,1,0,1,0,1,0,1,0,1 (see the number 3647).

15 376

• the only solution n

1012

of σ(n) = 2n + 31 (see the number 196).

15 613

• the only known number whose cube is the sum of a square and an eighth power:

here 15

6133

= 1 549

0342

+

338

(see the number 122).

15 727

• the smallest prime number that is preceded by 43 consecutive composite num-

bers; indeed, there are no primes between 15 683 and 15 727.

15 822

• the

20th

number n such that n ·

2n

− 1 is prime (see the number 115).

15 841

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

15 860

• the smallest number n 7 such that β∗(n) = β∗(n + 1) = β∗(n + 2), where

β∗(n) = β(n) − P (n) =

∑

p|n

pP (n)

p: here 15 860 = 22 · 5 · 13 · 61, 15 861 =

3 · 17 · 311 and 15 862 = 2 · 7 · 11 · 103, so that the common value of β∗(n + i) is

20; the sequence of numbers satisfying this property begins as follows: 15860,

46035, 101121, 167143, 414408, 526370, 593503, . . .

155;

if nk stands for the

smallest number n 8 such that β∗(n) = β∗(n + 1) = . . . = β∗(n + k − 1),

then n2 = 65, n3 = 15 860, n4 = 2 071 761 216, n5 ≤ 60 502 217 031 967 and

n6 ≤ 2 393 703 338 691 891 312 (compare with the number 89 460 294).

15 872

• the

14th

Granville number (see the number 126).

155One can prove that if the Generalized Twin Prime Conjecture is true, then, for each integer

k ≥ 2, equation β∗(n) = β∗(n + 1) = . . . = β∗(n + k − 1) has infinitely many solutions (see J.M. De

Koninck [46]).