Those Fascinating Numbers 25

77

• the largest number which cannot be written as the sum of positive integers

whose sum of reciprocals is equal to 1; thus

78 = 2 + 6 + 8 + 10 + 12 + 40 and

1

2

+

1

6

+

1

8

+

1

10

+

1

12

+

1

40

= 1

(R.L. Graham [95]);

• the smallest champion number whose smallest prime factor is 7: we say that n

is a champion number if D(n) D(m) for each positive integer m n, where

D(n) = τ (n!) − τ ((n − 1)!) (see A. Ivi´ c & C. Pomerance [112]).

78

• the smallest solution of σ2(n) = σ2(n + 13);

• the number of digits in the decimal expansion of the Fermat number

228

+ 1;

• the number of pseudoprime numbers in base 2 smaller than 107; we say that

an odd composite number n is a pseudoprime number in base 2 if 2n−1 ≡ 1

(mod n); see the number 245;

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

contain the digit 8; indeed, using a computer, one can verify that

278

= 302231454903657293676544

does not contain the digit 8, while each number

2k,

for k = 79, 80, . . . , 3000,

contains it (see the number 71);

79

• the smallest number which cannot be written as the sum of less than 19 fourth

powers: here 79 = 15 · 14 + 4 · 24;

• the second prime number p such that 31p−1 ≡ 1 (mod p2): the only prime

numbers p 232 satisfying this congruence are 7, 79, 6 451 and 2 806 861 (see

Ribenboim [169], p. 347).

80

• the smallest number n such that τ (n) τ (n + 1) τ (n + 2) τ (n + 3): here

10 5 4 2 (see the number 45);

• the second number for which the product of its proper divisors is a fourth

power, that is such that

d|n, dn

d =

a4:

here

2 · 4 · 5 · 8 · 10 · 16 · 20 · 40 = 40 960 000 =

804;

48 is the smallest number satisfying this property.