268 Jean-Marie De Koninck

711 828 (= 22 · 34 · 133)

• the smallest number n whose index of composition is ≥ 3 (namely λ(711828) ≈

3.09307) and which is the sum of two co-prime 3-powerful numbers: here n =

711 828 =

193

+

893.

712 800

• the tenth number n 1 such that φ(σ(n)) = n (see the number 128).

714 240 (= 29 · 32 · 5 · 31)

• the seventh Erd˝ os-Nicolas number (see the number 2 016).

716 035

• the fourth number n such that

n2

+ 2 is powerful: here

716

0352

+ 2 =

33

·

412

·

33612

(see the number 265).

720 720

• the largest number n that is divisible by each of the numbers ≤

n1/5,

namely

here the numbers 1, 2, 3, . . . , 14 (see N. Ozeki [157]);

• the 11th colossally abundant number (see the number 55 440).

721 970

• the smallest number n such that β(n) ≤ β(n + 1) ≤ . . . ≤ β(n + 9): here 146

927 1108 6091 10955 28884 90249 240662 360991 721979

(see the number 714).

729 551

• the smallest prime number q such that

∑

p≤q

p is divisible by 30 030 (= 2 · 3 ·

5 · 7 · 11 · 13): here this sum is equal to 20 488 357 890 (see the number 269).

729 752

• the tenth number n such that φ(n)σ(n) is a fourth power: here

φ(729752)σ(729752) =

8404

(see the number 170).