332 Jean-Marie De Koninck

594 839 010

• the smallest number which is not a sixth power, but which can be written as

the sum of the sixth powers of some of its prime factors: here 594 839 010 =

2 · 3 · 5 · 17 · 29 · 37 · 1087 = 26 + 56 + 296 (see the number 870); most likely, there

are infinitely many numbers satisfying this property: here are some of them:

7 315 846 275 =

56

+

376

+

416,

26 808 758 158 962 =

26

+

136

+

1736,

6 590 637 381 650 115 =

56

+

296

+

4336,

25 071 688 922 472 930 =

26

+

56

+

5416,

36 903 484 203 578 499 =

136

+

1016

+

5776,

458 900 605 733 751 282 =

26

+

3976

+

8776,

8 522 729 248 500 499 604 =

26

+

2296

+

475816,

924 192 866 401 605 860 540 =

26

+

56

+

136

+

296

+

31216,

471 757 284 187 831 225 961 144 274 =

26

+

136

+

279016,

1 403 603 924 704 633 779 104 023 266 =

26

+

57416

+

334616,

6 555 100 645 061 845 007 313 131 715 =

56

+

28976

+

432616,

19 005 354 284 570 728 948 848 506 180 =

26

+

56

+

116

+

5236

+

516596.

607 323 321 (= 32 · 115 · 419)

• the smallest number with an index of composition 2 which can be written as

the sum of two co-prime numbers each with an index of composition ≥ 6: we

have

607 323 321 =

32

·

115

· 419 =

25

·

58

+

296,

where

λ(32

·

115

· 419) ≈ 2.12123,

λ(25

·

58)

≈ 7.09691,

λ(296)

= 6.

608 892 570 (= 2 ·

32

· 5 ·

113

· 13 · 17 · 23)

• the smallest number which is not a prime power, but which is divisible by the

sum of the fourth powers of its prime factors, a sum which here is equal to

407286 = 2 ·

32

·

113

· 17 (see the number 378).

612 220 032 (=

27

·

314)

• the smallest number n 1 whose sum of digits is equal to

7

√

n: the only

other numbers n 1 satisfying this property are 10 460 353 203, 27 512 614 111,

52 523 350 144, 271 818 611 107, 1 174 711 139 837, 2 207 984 167 552 and

6 722 988 818 432.