318 Jean-Marie De Koninck

87 539 319

• the smallest number which can be written as the sum of two cubes in three dis-

tinct ways: 87 539 319 = 1673 +4363 = 2283 +4233 = 2553 +4143 (Leech, 1957):

the sequence of numbers satisfying this property begins as follows: 87 539 319,

119 824 488, 143 604 279, 175 959 000, 327 763 000, . . . ; see the number 1 729 as

well as R.K. Guy [101], D1.

87 699 842

• the smallest Niven number n such that n + 100 is also a Niven number, but

with no others in between (see the number 28 680).

88 593 477

• the largest number equal to the sum of the eighth powers of its digits (see the

number 24 678 050).

88 612 802

• the smallest number which can be written as the sum of two, three and four

distinct fourth powers: indeed, 88 612 802 = 174 + 974 = 214 + 804 + 834 =

254 + 264 + 604 + 934.

89 002 914

• the tenth number n such that β(n)|β(n +1) and β(n +1)|β(n +2), here 138|966

and 966|966 (see the number 225 504).

89 351 671

• the only prime number p

232

such that

66p−1

≡ 1 (mod

p2)

(P.L. Montgo-

mery [143]).

89 397 016

• the 15th dihedral perfect number (see the number 130).

89 460 294

• the smallest number n (and the only one 1011) such that (∗) β(n) = β(n +

1) = β(n + 2); here n = 2 · 3 · 7 · 11 · 23 · 8419, n + 1 = 5 · 4201 · 4259 and

n + 2 =

23

· 31 · 43 · 8389, so that β(n) = β(n + 1) = β(n + 2) = 8465; although

n = 89460294 is the only known solution of (∗), one may be lead to believe that

(∗) has infinitely many

solutions195

(see the number 417 162 for the analogue

problem when one replaces β(n) by B(n), as well as the number 15 860 for the

similar problem when β(n) is replaced by β∗(n)).

195In fact, if C(x) stands for the number of solutions n ≤ x of (∗), Carl Pomerance (see J.M. De

Koninck [46]) provided a heuristic argument suggesting that not only C(x) → ∞ as x → ∞, but

that, given any real number ε 0, C(x) x1−ε for all x suﬃciently large.