Those Fascinating Numbers 129
1 444 (= 382)
the smallest four digit perfect square which has only two distinct digits: Hito-
tumatu conjectured that, besides the numbers of the form 102n, 4 · 102n and
9 · 102n, there is only a finite number of perfect squares with only two distinct
digits (R.K. Guy [101], F24);
the second number n 1 such that σ(n) and σ2(n) have the same prime factors,
namely the primes 3, 7 and 127 (see the number 180);
the smallest perfect square whose last three digits are 444: the following129 one
is 213 444.
1 445
the fourth Ap´ ery number: the sequence of Ap´ ery numbers a0, a1, a2, . . . is de-
fined by
an =
n
k=0
n
k
n + k
k
2
=
n
k=0
((n +
k)!)2
(k!)4((n k)!)2
(n = 0, 1, 2, 3, . . .),
and begins as follows: 1, 5, 73, 1445, 33 001, 819 005, 21 460 825, 584 307 365,
16 367 912 425, 468 690 849 005, 13 657 436 403 073, . . .
1 458
the third number n such that φ(n), φ(n + 1) and φ(n + 2) have the same prime
factors, namely here 2 and 3: the sequence of numbers satisfying this property
begins as follows: 35, 36, 1 458, 3 456, 16 921, . . . (see the numbers 266 401 and
3 777);
the sixth number m such that

n≤m
φ(n) is a perfect square: here

n≤1458
φ(n) = 646 416 =
8042;
if mk stands for the
kth
number such that
sk =

n≤mk
φ(n) is a perfect square, here are the values of mk, for 1 k 12,
along with the corresponding values sk:
k mk sk

sk
1 1 1 1
2 3 4 2
3 14 64 8
4 32 324 18
5 54 900 30
6 1458 646416 804
k mk sk

sk
7 3765 4309776 2076
8 5343 8678916 2946
9 10342 32512804 5702
10 57918 1019652624 31932
11 72432 1594724356 39934
12 134072 5463870724 73918
129One
can prove that all numbers of the form (500r ±
38)2,
r = 0, 1, 2, . . ., satisfy this property
(see M. Gardner [87], p. 270).
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