260 Jean-Marie De Koninck

432 000

• the tenth number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).

435 708 (=

22

·

32

·

72

· 13 · 19)

• the fourth solution of

σ(n)

n

=

10

3

(see the number 1 080).

440 312

• the number of twin prime pairs

108

(see the number 1 224).

441 461

• the

13th

and largest prime number q for which the value of the corresponding

sum

∑

p≤q

p uses each of the digits 0,1,2,. . . ,9 once and only once: in this case,

∑

p≤441461

p = 7 803 615 924 (see the number 155 863).

453 962

• the eighth number n divisible by a square 1 and such that δ(n +1)−δ(n) = 1

(see the number 49).

455 226

• the smallest number which is equal to the sum of the seventh powers of its digits

added to the product of its digits: the only numbers satisfying this property

are 455 226, 3 653 786, 4 210 818, 7 774 369 and 9 800 817.

461 890

• the smallest integer n such that ω(n), ω(n + 1), . . . , ω(n + 5) are all distinct,

namely in this case with the values 6, 1, 4, 2, 3 and 5 (see the number 417).

465 124

• the eighth powerful number n such that n + 1 is also powerful: here 465 124 =

22 · 112 · 312 and 465 125 = 53 · 612 (see the number 288).