Those Fascinating Numbers 31

begins as follows: 1, 6, 92, 406, 1 549, 5 361, 12 546, 41 908, 141 121, . . . ; it is

well known that the function M(x) changes sign infinitely often as x → ∞ (see

E. Grosswald [98]), which implies in particular that the number mk does indeed

exist for each k ≥ 1.

93

• the largest known number k such that the decimal expansion of 2k does not

contain the digit 6; indeed, using a computer, one can verify that

293

= 9903520314283042199192993792

does not contain the digit 6, while each number

2k,

for k = 94, 95, . . . , 3000,

contains it (see the number 71);

• the smallest number 1 whose sum of divisors is a seventh power: σ(93) =

27.

94

• the seventh number whose sum of divisors is a perfect square: σ(94) = 112 the

sequence of numbers satisfying this property begins as follows: 1, 3, 22, 66, 70,

81, 94, 115, 119, 170, 210, 214, . . .

96

• the smallest number n such that each number m n can be written as the

sum of distinct elements from the set {ppn : n = 1, 2, . . .} = {3, 5, 11, 17, . . .}

(see R.E. Dressler & S.T. Packer [69]);

• the smallest number m such that equation σ(x) = m has exactly four solutions,

namely 42, 62, 69 and 77;

• the smallest number n 1 such that

γ(n)2|σ(n):

the sequence of numbers satis-

fying this property begins as follows: 1, 96, 864, 1080, 1782, 6144, 7128, . . .

36;

if nk stands

for37

the smallest number n for which

σ(n)/γ(n)k

is an integer,

then n1 = 6, n2 = 96, n3 = 3 538 944 and n4 ≤ 19 698 744 770 118 549 504 (see

the number 1 782).

97

• the smallest prime number preceded by exactly seven consecutive composite

numbers; indeed, there are no prime numbers between 89 and 97; if qk stands for

the smallest prime number that is preceded by exactly k consecutive composite

numbers, then q1 = 5, q3 = 11, q5 = 29, q7 = 97, q9 = 149, q11 = 211,

q13 = 127, q15 = 1 847, q17 = 541, q19 = 907 and q21 = 1 151;

• the largest two digit prime number.

36It

is easy to see that there exist infinitely many numbers satisfying this property, namely all

the numbers n =

2α3β

, where α + 1 is a multiple of 6 and β ≥ 1 is odd.

37By

examining the numbers of the form n =

2α3β

with an appropriate choice of α and β, one

can easily discover infinitely many numbers n for which γ(n)k|σ(n) for any fixed number k.