8 Jean-Marie De Koninck
2-hyperperfect 21 = 3 · 7
2 133 = 33 · 79
19 521 = 34 · 241
176 661 = 35 · 727
129 127 041 =
38
· 19 681
328 256 967 373 616 371 221 =
321
· 31381059607
3-hyperperfect 325 = 52 · 13
4-hyperperfect 1 950 625 =
54
· 3 121
1 220 640 625 =
56
· 78 121
186 264 514 898 681 640 625 =
514
· 30 517 578 121
6-hyperperfect 301 = 7 · 43
16 513 =
72
· 337
60 110 701 =
72
· 383 · 3203
1 977 225 901 =
75
· 117 643
2 733 834 545 701 =
74
· 30893 · 36857
232 630 479 398 401 =
78
· 40353601
10-hyperperfect 159 841 = 112 · 1 321
11-hyperperfect 10 693 = 172 · 37
12-hyperperfect 697 = 17 · 41
2 041 = 13 · 157
1 570 153 = 13 · 269 · 449
62 722 153 =
133
· 28 549
10 604 156 641 =
134
· 371 281
13 544 168 521 =
132
· 2347 · 34147
1 792 155 938 521 =
135
· 4 826 797
the number of two digit prime numbers; if we let C(k) stand for the num-
ber of k digit prime numbers, then C(1) = 4, C(2) = 21, C(3) = 143,
C(4) = 1 061, C(5) = 8 363, C(6) = 68 906, C(7) = 586 081, C(8) = 5 096 876,
C(9) = 45 086 079, C(10) = 404 204 977, C(11) = 3 663 002 302 and C(12) =
33 489 857 205.
22
the smallest Smith number: a composite number is said to be a Smith number
if the sum of its digits is equal to the sum of the digits of its distinct prime
factors: here 22 = 2 · 11 and 2 + 2 = 4 = 2 + 1 + 1 (see U. Dudley [72]).
23
the prime number which appears the most often as the fifth prime factor of an
integer (see the number 199);
one of the two numbers (the other one being 239) which cannot be written as
the sum of less than nine cubes (of non negative integers): here 23 = 2·23 +7·13
(L.E. Dickson [66]);
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