132 Jean-Marie De Koninck
1 547
the number of Carmichael numbers
1010
(see the number 646).
1 549
the number of integer zeros of the function M(x) :=
n≤x
µ(n) located in the
interval [1, 100 000] (see the number 92).
1 560
the smallest number n such that σ(n) = 7!; if nk stands for the smallest number
n such that σ(n) = k!, then n3 = 5, n4 = 14, n5 = 54, n6 = 264, n7 = 1 560,
n8 = 10 290, n9 = 97 440, n10 = 876 960, n11 = 10 263 240 and n12 = 11 289 564
(see the number 779 for the analogue question for equation φ(n) = k!).
1 568 (=
25
·
72)
the second number n having at least two distinct prime factors and such that
B1(n) =
β(n)2:
here
25
+
72
= (2 +
7)2
(see the number 144).
1 575
the second of the existing eight primitive non deficient numbers (see the number
945).
1 597
the seventh prime Fibonacci number (see the number 89).
1 599
the smallest solution of τ (n+1)−τ (n) = 13; the sequence of numbers satisfying
this property begins as follows: 1599, 2209, 2915, 5329, 6889, 7743, 9999, . . .
1 600
the second perfect square which is a Smith number (see the number 22): 1 600 =
26
·
52
and 1 + 6 = 7 = 2 + 5; the smallest with this property is 361.
1 615
the second composite number n such that σ(n + 8) = σ(n) + 8; the sequence
of numbers satisfying this equation begins as follows: 27, 1 615, 1 885, 218 984,
4 218 475, . . . (compare with the number 305 635 357).
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