Those Fascinating Numbers 347

4 931 691 075

• the 13th powerful number n such that n+1 is also powerful: here 4 931 691 075 =

35 · 52 · 172 · 532 and 4 931 691 076 = 22 · 132 · 372 · 732 (see the number 288).

5 391 411 025 (= 52 · 7 · 11 · 13 · 17 · 19 · 23 · 29)

• the smallest odd abundant number which is not a multiple of 3.

5 394 826 801 (= 7 · 13 · 17 · 23 · 31 · 67 · 73)

• the smallest Carmichael number which is the product of seven prime numbers

(see the number 41 041).

5 425 069 447

• the 14th powerful number n such that n+1 is also powerful: here 5 425 069 447 =

73 · 412 · 972 and 5 425 069 448 = 23 · 260412 (see the number 288).

5 471 312 310

• the smallest number n which allows the sum

i≤n

1

i

to exceed 23 (see the number

83).

5 745 705 602

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

powers in eight distinct ways:

5 745 705 602 =

34

+

2304

+

2334

=

254

+

2184

+

2434

=

434

+

2074

+

2504

=

584

+

1974

+

2554

=

854

+

1774

+

2624

=

904

+

1734

+

2634

=

1024

+

1634

+

2654

=

1224

+

1454

+

2674

(see the number 6 578).

6 058 655 748

• the smallest number which can be written as the sum of the cubes of two prime

numbers in two distinct ways: 6 058 655 748 = 613 + 18233 = 10493 + 16993;

the sequence of numbers satisfying this property begins as follows: 6058655748,

6507811154, 12906787894, 20593712932, 140253191624, 293833825922, . . . (for

fourth powers, see the number 3 262 811 042)