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 =

2α

· 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 2α · 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.