Those Fascinating Numbers 93
548
the smallest number r which has the property that each number can be written
as x1 9 +x2 9 +. . .+xr, 9 where the xi’s are non negative integers (see the number 4).
550
the smallest solution of σ(n) = 2n + 16; the sequence of numbers satisfying
this equation begins as follows: 550, 748, 1 504, 7 192, 7 912, 10 792, 17 272,
30 592,. . .
98
557
the largest prime number n such that π(n)
n
log n
+
n
log2
n
, namely the first
two terms of the asymptotic expansion of Li(n): here π(557) = 102 while
n
log n
+
n
log2
n
n=557
102.031.
560
the second number n such that φ(n) + σ(n) = 3n (see the number 312).
561 (= 3 · 11 · 17)
the smallest Carmichael number: a composite number n is called a Carmichael
number if bn−1 1 (mod n) for all numbers b: the infinite sequence (see the
number 646) of Carmichael numbers begins as follows: 561, 1105, 1729, 2465,
2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973,
75361, . . .
563
the largest known Wilson prime (Goldberg, 1953): see the number 5;
the second prime number q such that

p≤q
p is a multiple of 100 (here this sum
is equal to 25 800); the sequence of numbers satisfying this property begins as
follows: 23, 563, 937, 2099, 3371, 5407, 6977, . . . ;
the smallest number n such that θ(x) x(1 1/(2 log x)) for all real numbers
x n (here θ(x) =

p≤x
log p).
98It
is easy to show that each number n =

· p, where α is a positive integer such that
p = 2α+1 17 is prime, is a solution of σ(n) = 2n + 16: it is the case in particular when
α = 5, 7, 11, 15, 17, 19, 21, 23, 31, 35, 41, 43, 95; the solutions corresponding to α = 5 and α = 7
are included in the above list. It is also possible to identify the solutions n of the form · p · q, with
p q primes and α positive integers. We thus obtain the above solutions n =
22
· 11 · 17 = 748,
n = 23 · 29 · 41 = 7 192, n = 23 · 23 · 43 = 7 912, n = 23 · 19 · 71 = 10 792 and n = 23 · 17 · 127 = 17 272.
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