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).