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]).
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