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.
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