Those Fascinating Numbers 189
12 758
the largest number which is not the sum of distinct cubes (see Journal of Recre-
ational Mathematics 20 (1988), p. 316).
12 769
the smallest powerful number which can be written as the sum of two co-prime
4-powerful numbers: 12 769 = 2 401 + 10 368, that is
1132
=
74
+
27
·
34;
there
are at least four other pairs of co-prime 4-powerful numbers {a, b} such that
a + b is powerful; these are
35
+
114
=
22
·
612,
25
·
194
+
74
·
175
=
372
·
15792,
74
·
1674
+
25
·
34
·
54
·
116
=
172
·
1280332,
24
·
234
·
374
+
36
·
114
·
474
=
52
·
3132
·
49692;
the seventh powerful number which can be written as the sum of two co-prime
3-powerful numbers = 1: 12 769 = 2 401 + 10 368, that is
1132
=
74
+
27
·
34
(see the number 841).
12 819
the seventh number n such that 2n 7 is prime (see the number 39).
12 853
the smallest number n such that τ (n) τ (n + 1) . . . τ (n + 6) (as well as
the smallest such that τ (n) τ (n + 1) . . . τ (n + 7)): here 2 4 8
8 8 8 8 12 (see the number 241).
12 875
the only solution y of the diophantine equation x(x+1)(x+2) . . . (x+5) =
y2

25: the only solution of this equation is indeed (x, y) = (21, 12875) (L.E. Mattics
[132]); see the footnote tied to the number 142.
12 996
the third number such that σ(n) and σ2(n) have the same prime factors, namely
the primes 3, 7, 13 and 127 (see the number 180).
13 033
the
17th
prime number pk such that p1p2 . . . pk 1 is prime (see the number
317).
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