Those Fascinating Numbers 267

673 663

• the third composite number n such that 2n−4 ≡ 1 (mod n); see the number

40 369.

675 214

• the smallest number n which allows the sum

i≤n

1

i

to exceed 14 (see the num-

ber 83).

678 570

• the

11th

Bell number (see the number 52).

703 125

• the smallest number whose index of composition is ≥ 4 and which can be

written as the sum of two co-prime numbers each with an index of composition

≥ 3: indeed,

703 125 = 2 197 + 700 928,

32

·

57

=

133

+

29

·

372,

with

λ(32

·

57)

≈ 4.97158,

λ(133)

= 3 and

λ(29

·

372)

≈ 3.12731; the sequence of

numbers satisfying this property begins as follows: 703 125, 720 896, 3 418 801,

6 815 744, . . . 185

706 063

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

φ(706063)σ(706063) = 8404 (see the number 170).

706 237

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

φ(706237)σ(706237) = 8404 (see the number 170).

709 838

• the fifth number n such that

∑

p≤pn

p is a perfect square: here we have

∑

p≤p709838

p =

19163572

(see the number 2 474).

185One

can easily show that if the abc Conjecture is true, then there is only a finite number of

numbers with this property.