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]).
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