326 Jean-Marie De Koninck

224 376 048

• the number of twin prime pairs 1011 (see the number 1 224).

240 589 440 (=

27

·

35

· 5 · 7 · 13 · 17)

• the second solution of

σ(n)

n

=

14

3

(see the number 208 565 280).

248 832 000

• the smallest number 2 which is equal to the product of the factorials of its

digits in base 7: 248 832 000= [6, 1, 1, 1, 0, 1, 5, 5, 0, 4]7 = 6! · 1! · 1! · 1! · 0! · 1! · 5! ·

5! · 0! · 4!; the only known numbers satisfying this property are 1, 2, 248 832 000,

1 433 272 329, 15 407 021 574 586 368 000,

1 831 607 359 566 125 048 834 492 989 440 000 000 000 and

273 457 513 497 334 816 890 950 735 729 000 448 000 000 000 000 (see the number

17 280 for the list of the smallest numbers with this property in different bases).

257 400 763

• the sixth number which can be written in two distinct ways as the sum of

two co-prime numbers each with an index of composition ≥ 5: 257 400 763 =

53 · 77 + 25 · 136 = 212 · 52 + 37 · 76, each of these last four numbers having as

index of composition 5.18928, 5.78725, 5.0103 and 6.36085 respectively (see the

number 371 549).

258 474 216

• the largest triangular number which is the product (Anglin [6], p. 21) of three

consecutive numbers:

258 474 216 = 636 · 637 · 638 =

22 736 · 22 737

2

.

263 215 633

• the number n which allows the sum

m≤n

ω(m)=1

1

m

to exceed 4 (see the number

1 307).

272 400 600

• the smallest number n which allows the sum

i≤n

1

i

to exceed 20 (see the number

83).