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.
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