Those Fascinating Numbers 317

81 675 000 (= 23 · 33 · 55 · 122)

• the number, amongst all those

108,

whose index of composition is the nearest

to the number π: λ(81675000) ≈ 3.14157 (see the number 7 826 354 460 for a

better approximation, and see the number 1 156 000 for the similar question

regarding the number e).

82 564 350

• the second number n such that

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

5

4

:

here

min(λ(n), λ(n + 1), λ(n + 2), λ(n + 3), λ(n + 4))

≈ min(1.28177, 1.36191, 1.36271, 1.4313, 1.35013) = 1.28177

(see the number 14 018 750).

82 623 911

• the

15th

solution of σ2(n) = σ2(n + 2) (see the number 1 079).

83 623 935 (= 3 · 5 · 17 · 353 · 929)

• the smallest number n such that φ(n)|(n + 1) and which is not exclusively the

product of Fermat primes (see the number 65 535).

85 016 574 (= 2 ·

33

· 29 · 233)

• the smallest number n such194 that min(λ(n), λ(n + 1), λ(n + 2)) 1.72: here

with n = 85 016 574, we have λ(n) ≈ 1.72085, λ(n+1) ≈ 1.80738 and λ(n+2) ≈

1.97442; see also the number 9 077 457 159 999 998.

85 864 769

• the smallest prime number q1 such that each number qi = 2qi−1 + 1 is prime

for i = 2, 3, . . . , 9: such a sequence of prime numbers is called a Cunningham

chain (see the number 1 122 659).

194One

can prove that if the abc Conjecture is true, then, for any fixed ε 0, there is only a finite

number of numbers n such that min(λ(n), λ(n+1), λ(n+2))

3

2

+ε, while one can prove without any

condition that there exist infinitely many numbers n such that min(λ(n), λ(n +1), λ(n +2))

3

2

− ε

(see J.M. De Koninck & N. Doyon [48]).