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.
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