80 Jean-Marie De Koninck
378
the smallest number which is not a cube, but which can be written as the sum
of the cubes of its prime factors: 378 = 2 ·
33
· 7 =
23
+
33
+
73;
there exist at
least seven other
numbers84
satisfying this property, namely:
2 548 =
22
·
72
· 13 =
23
+
73
+
133,
2 836 295 = 5 · 7 · 11 · 53 · 139 =
53
+
73
+
113
+
533
+
1393,
4 473 671 462 = 2 · 13 · 179 · 593 · 1621 =
23
+
133
+
1793
+
5933
+
16213,
23 040 925 705 = 5 · 7 · 167 · 1453 · 2713 =
53
+
73
+
1673
+
14533
+
27133,
13 579 716 377 989 = 19 · 157 · 173 · 1103 · 23857
=
193
+
1573
+
1733
+
11033
+
238573,
21 467 102 506 955 = 5 ·
73
· 313 · 1439 · 27791 =
53
+
73
+
3133
+
14393
+
277913,
119 429 556 097 859 = 7 · 53 · 937 · 6983 · 49199 =
73
+
533
+
9373
+
69833
+
491993;
the smallest number which is not a prime power, but which is divisible by the
sum of the cubes of its prime factors; given a positive integer k, if we denote
by nk the smallest number n such that ω(n) = 1 and such that βk(n)|n, where
βk(n) =

p|n
pk, we have the following table:
k nk βk (n)
1 30 = 2 · 3 · 5 10 = 2 · 5
2 46 206 = 2 ·
32
· 17 · 151 23 103 =
32
· 17 · 151
3 378 = 2 ·
33
· 7 378 = 2 ·
33
· 7
4 608 892 570 = 2 ·
32
· 5 ·
113
· 13 · 17 · 23 407 286 = 2 ·
32
·
113
· 17
5 292 353 065 550 39 352 950 = 2 ·
32
·
52
· 7 · 13 ·
312
= 2 ·
32
·
52
· 7 · 13 · 17 · 19 · 23 ·
312
6 539 501 733 634 012 578 19 184 230 593 =
37
· 11 ·
192
·
472
= 2 ·
37
· 11 · 13 ·
192
· 23 · 31 · 37 · 41 ·
472
As for the value of n7, one can claim that
n7 n = 1 149 039 082 866 174 511 355 807 661 240,
since n =
23
· 3 · 5 ·
72
·
174
· 23 · 37 · 43 · 47 · 53 · 61 · 67 · 73 · 79 · 97 · 103 · 109 and
β7(n) =
23
·
72
·
174
· 37 · 47 · 73 · 103.
(see also the number 99 528).
379
the seventh prime number pk such that p1p2 . . . pk + 1 is prime: the only known
prime numbers satisfying this property are 2, 3, 5, 7, 11, 31, 379, 1 019, 1 021,
2 657, 3 229, 4 547, 4 787, 11 549, 13 649, 18 523, 23 801, 24 029, 42 209, 145 823,
366 439 and 392 113 (see R.K. Guy [101], A2 and C. Caldwell [29]; see also the
number 211);
84If
stands for the set of numbers which can be written as the sum of the
αth
powers of their
prime factors, then J.M. De Koninck & F. Luca [54] proved that the only elements of S3 having
exactly three prime factors are 378 and 2 548. On the other hand, it is also possible that #Sα = 0
for each α 2, α = 3.
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