82 Jean-Marie De Koninck

k ρ(k)

2 2

4 13

5 5

7 7

8 17

10 19

11 29

13 67

14 59

16 79

17 89

k ρ(k)

19 199

20 389

22 499

23 599

25 997

26 1889

28 1999

29 2999

31 4999

32 6899

34 17989

k ρ(k)

35 8999

37 29989

38 39989

40 49999

41 59999

43 79999

44 98999

46 199999

47 389999

49 598999

50 599999

k ρ(k)

52 799999

53 989999

55 2998999

56 2999999

58 4999999

59 6999899

61 8989999

62 9899999

64 19999999

65 29999999

67 59899999

k ρ(k)

68 59999999

70 189997999

71 89999999

73 289999999

74 389999999

76 689899999

77 699899999

79 799999999

80 998999999

82 2999899999

83 3999998999

390 (= 2 · 3 · 5 · 13)

• the fifth ideal number: an r digit number is said to be an ideal number if its

jth factor, for each 1 ≤ j ≤ r, is the one that appears the most often as the

jth prime factor of an integer (see the number 199): the first 20 ideal numbers

are: 2, 6, 30, 42, 390, 546, 8970, 12558, 421590, 590226, 47639670, 66695538,

9480294330, 13272412062, 682923295390, 3756092613546, 1252925178947130,

1754095250525982, 1111344633726104310 and 1555882487216546034;

• the smallest number which cannot be written as the sum of seven non zero

distinct squares (R.K. Guy [101], C20).

392

• the only solution n

1012

of σ(n) = 2n + 71 (see the number 196).

396

• the third number n = [d1, d2, . . . , dr] such that (d1 +1)·(d2 +2)·. . .·(dr +r) = n;

the sequence of numbers satisfying this property is infinite and includes the

numbers 16, 27, 396, 117 729 612 000, 266 744 016 000, 722 905 497 600,

1 234 057 144 320 and 16 231 012 761 600.

399 (= 3 · 7 · 19)

• the

14th

number n such that n! + 1 is prime (see the number 116);

• the smallest solution of τ (n + 1) − τ (n) = 7; the sequence of numbers satisfying

this equation begins as follows: 399, 783, 2703, 3249, 4623, 11024, 15129, 16383,

17689, . . . ; if nk stands for the smallest number n such that τ (n+1)−τ (n) = k,

then the sequence (nk)k≥1 begins as follows: 1, 5, 49, 11, 35, 23, 399, 47, 1849,

59, 143, 119, 1599, 167, 575, 179, 1295, 239, 4355, 629, 2303, 359, 899, 959,

9215, 1007, 39999, 719, 20735, 839, . . . ;

• the largest number of the form 8k + 7 which can be written as the sum of

exactly three powerful numbers in only one way: here 399 = 49 + 125 + 225

(see the number 118);