Those Fascinating Numbers 53

183

• the smallest Sastry number: we say that n is a Sastry number if when we

concatenate it with its successor we obtain a perfect square: here 183184 =

4282;

the sequence of Sastry numbers begins as follows: 183, 328, 528, 715,

6099, 13224, 40495, 106755, . . . ; one can prove that there exist infinitely Sastry

numbers64.

184 (= 23 · 23)

• the sixth dihedral perfect number (see the number 130);

• the

100th

number having exactly two distinct prime factors; if nk stands for the

10k-th

number having exactly two distinct prime factors, then n1 = 24, n2 =

184, n3 = 2 102, n4 = 26 608, n5 = 322 033, n6 = 3 741 154 and n7 = 42 314 023

(see the table accompanying the number 455).

185

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

starting with n, the values 1, −1, 1, −1, 1, −1; the sequence of numbers satis-

fying this property begins as follows: 58, 185, 194, 274, 287, 342, 344, 382, 493,

. . .

188

• the largest number which cannot be written as the sum of less than five dis-

tinct squares: 124 and 188 are the only two numbers satisfying this property

(R.K. Guy [101], C20);

• the third solution of σ(n) = σ(n + 7) (see the number 62).

189

• the fourth solution of

φ(n)

n

=

4

7

; the sequence of numbers satisfying this equa-

tion begins as follows: 21, 63, 147, 189, 441, 567, 1029, 1323, 1701, . . . 65

190

• the fifth pseudoprime in base 11: the ten smallest are 10, 15, 70, 133, 190, 259,

305, 481, 645 and 703.

64In

fact, one can prove that there exists a Sastry number with r digits if and only if

10r

+ 1 is

not a prime number (see F. Luca [126]).

65It

is easy to prove that n is a solution of

φ(n)

n

=

4

7

if and only if n =

3k

· 7 for certain integers

k ≥ 1 and ≥ 1.