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).