232 Jean-Marie De Koninck
n
11
+
22
+
33
+ . . . +
nn
51 24 · 3 · 11 · 154639601191 · 315009604207711754599670873
·47750329195302200576715751641087454974565302619
52 24 · 71 · 330719 · 390161 · 37397876028774143
·31358805941069574823971404439468355888575789769408711369171
53
32
· 5 · 83 · 227 · 943891331 · 961780171211 · 2309072369999773039
·13801415901334152246975826518180877780071327199
54
32
· 89 · 233 · 1297 · 17781070313573787458259405149
·828655068552707241561712640063455340626001805902851628449
55
22
· 347 · C93
56
22
· 756667 · P92
57 23 · 11003423 · 514573761161 · 45616193451775370186003
·376654681044364381912063667 · 5469384461780255260307037405181
58
32
· 18199 · 482387 · C92
59
23
· 13 · 2531 · 4705684631115340784267 · P78
60
23
· 7 · 193 · 130140589 · C95
88 916
the second number n such that n2 +n+1 is powerful (here 88 9162 +88 916+1 =
7 906 143 973 = 72 · 133 · 2712); there exist infinitely many numbers satisfying
this property168; the only other number n 107 satisfying this property169 is
n = 18 (here 182 + 18 + 1 = 73).
88 935 (= 3 · 5 ·
72
·
112)
the smallest odd number n having four prime factors and such that γ(n)|φ(n)
(see the numbers 147 and 3 675).
90 625
the only five digit automorphic number: 90
6252
= 8 212 890 625 (see the num-
ber 76).
90 825
the ninth number n 1 such that n ·
2n
+ 1 is prime (see the number 141).
168Indeed,
we only need to consider equation
n2
+n+1 =
343x2,
that is
(2n+1)2 −343(2x)2
= −3,
which has infinitely many solutions (see the number 37).
169It
is easy to see that a number n such that
n2
+ n + 1 is powerful cannot be of the form 3k + 1,
the reason being that for such a number n, we have n2 + n + 1 3 (mod 9), so that 3 n2 + n + 1.
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