Those Fascinating Numbers 189

12 758

• the largest number which is not the sum of distinct cubes (see Journal of Recre-

ational Mathematics 20 (1988), p. 316).

12 769

• the smallest powerful number which can be written as the sum of two co-prime

4-powerful numbers: 12 769 = 2 401 + 10 368, that is

1132

=

74

+

27

·

34;

there

are at least four other pairs of co-prime 4-powerful numbers {a, b} such that

a + b is powerful; these are

35

+

114

=

22

·

612,

25

·

194

+

74

·

175

=

372

·

15792,

74

·

1674

+

25

·

34

·

54

·

116

=

172

·

1280332,

24

·

234

·

374

+

36

·

114

·

474

=

52

·

3132

·

49692;

• the seventh powerful number which can be written as the sum of two co-prime

3-powerful numbers = 1: 12 769 = 2 401 + 10 368, that is

1132

=

74

+

27

·

34

(see the number 841).

12 819

• the seventh number n such that 2n − 7 is prime (see the number 39).

12 853

• the smallest number n such that τ (n) ≤ τ (n + 1) ≤ . . . ≤ τ (n + 6) (as well as

the smallest such that τ (n) ≤ τ (n + 1) ≤ . . . ≤ τ (n + 7)): here 2 4 8 ≤

8 ≤ 8 ≤ 8 ≤ 8 12 (see the number 241).

12 875

• the only solution y of the diophantine equation x(x+1)(x+2) . . . (x+5) =

y2

−

25: the only solution of this equation is indeed (x, y) = (21, 12875) (L.E. Mattics

[132]); see the footnote tied to the number 142.

12 996

• the third number such that σ(n) and σ2(n) have the same prime factors, namely

the primes 3, 7, 13 and 127 (see the number 180).

13 033

• the

17th

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

317).