Those Fascinating Numbers 225

69 984

• the seventh number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).

70 226

• the second even number n such that n2 −1 is powerful: here 70 2262 −1 = 35 ·52·

172 ·532; the smallest such number is n = 26; the sequence of numbers satisfying

this property begins as follows: 26, 70 226, 130 576 328, 189 750 626,. . . and is

infinite165 (see also the number 485).

70 841

• the largest prime p such that π(p)

p

log p

+

p

log2

p

+

2p

log3

p

, namely the first

three terms of the asymptotic expansion of Li(p): here π(70 841) = 7012 while

p

log p

+

p

log2

p

+

2p

log3

p

p=70841

≈ 7012.77.

71 199

• the largest solution n 109 of γ(n + 1) − γ(n) = 11: the others are 20, 27, 288

and 675 (see the number 98).

71 825 (= 53 · 13 · 17)

• the smallest number which can be written as the sum of two squares in nine

distinct ways, namely 71 825 = 12 + 2682 = 402 + 2652 = 652 + 2602 = 762 +

2572

=

1042

+

2472

=

1272

+

2362

=

1602

+

2152

=

1692

+

2082

=

1882

+

1912

(see the number 50).

72 105 (= 3 · 5 · 11 · 19 · 23)

• the smallest odd number which is not a prime power, but which is divisible by

the sum of the squares of its prime factors (see the number 46 206).

165Indeed, one can see, through equation x2 − 27y2 = 1, that the numbers xr defined by relation

(26+15

√

3)2r−1

= xr +yr

√

3 are such numbers: one thus obtains the numbers x1 = 26, x2 = 70 226,

x3 = 189 750 626 and x4 = 512 706 121 226.