172 Jean-Marie De Koninck

6 397

• the prime number which appears the most often as the

15th

prime factor of an

integer (see the number 199).

6 399

• the second solution of τ (n + 1) − τ (n) = 17 (see the number 1 295).

6 435

• the 11th odd abundant number (see the number 945).

6 451

• the third prime number p such that

31p−1

≡ 1 (mod

p2)

(see the number 79).

6 474

• the third solution of σ2(n) = σ2(n + 12) (see the number 864).

6 489

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 17 (see the

number 177).

6 552 (=

23

·

32

· 7 · 13)

• the third solution of

σ(n)

n

=

10

3

(see the number 1 080).

6 569

• the

16th

prime number pk such that p1p2 . . . pk − 1 is prime (see the number

317).

6 578

• the smallest number which can be written as the sum of three distinct fourth

powers in two distinct ways: 6 578 =

14

+

24

+

94

=

34

+

74

+

84

(Martin,

1876); for the case when the condition “distinct” is not required, see the num-

ber 2 673; if nk stands for the smallest number n which can be written as

the sum of three distinct fourth powers in k distinct ways, then n2 = 6 578,

n3 = 811 538, n4 = 5 978 883, n5 = n6 = 292 965 218, n7 = 779 888 018 and

n8 = 5 745 705 602; for the analogue question with the fifth powers, see the

number 1 375 298 099; for the sixth powers, see the number 160 426 514;