288 Jean-Marie De Koninck

4 190 208

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

(see the number 48).

4 210 818

• the second number n 1 which can be written as the sum of the seventh

powers of its digits (see the number 1 741 725).

4 213 597

• the 12th Bell number (see the number 52).

4 218 475

• the fifth composite number n such that σ(n + 8) = σ(n) + 8 (see the number

1 615).

4 233 355

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

added to the product of its digits; the only other numbers with this property188

are 54 617 522, 146 511 208 and 874 917 848.

4 289 592

• the smallest number n such that ω(n)+ω(n+1)+ω(n+2)+ω(n+3) = 19: here

4 289 592 = 23·3·19·23·409, 4 289 593 = 7·11·17·29·113, 4 289 594 = 2·31·43·1609

and 4 289 595 = 3 · 5 · 37 · 59 · 131 (see the number 987).

4 324 320

• the

13th

colossally abundant number (see the number 55 440).

4 325 170

• the smallest number n such that P (n + i) ≤

√

n + i for i = 0, 1, 2, 3, 4, 5, 6, 7,

8, 9; the largest prime factors of these 10 numbers are respectively 701, 1033,

149, 1087, 1069, 409, 181, 331, 71 and 977, all smaller than

√

4325179 ≈ 2079

(see the number 1 518).

188One

can argue, as in the footnote tied to the number 1 324, that any such number can have at

most ten digits; thus using a computer, one only needs to check all numbers below 1010.