Those Fascinating Numbers 47

• the smallest solution of σ(n) = σ(n + 38); the sequence of numbers satisfying

this equation begins as follows: 147, 246, 249, 460, 627, 715, 716, 1026, 1509,

2427, 4857, 5396, 6815, 6954, 8672, 9447, 9714, 9752,. . . ;

• the second composite number n such that σ(n + 6) = σ(n) + 6 (see the number

104).

152

• the second solution of γ(n + 1) − γ(n) = 13: the only solutions n 109 of this

equation are 18, 152 and 3 024 (see the footnote tied to the number 98).

153

• the second number which can be written as the sum of the cubes of its digits:

153 = 13 + 53 + 33; the others are 1, 370, 371 and 407;

• the smallest narcissistic number of more than one digit (see the number 88);

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

contain the digit 3; indeed, using a computer, one can verify that

2153

= 11417981541647679048466287755595961091061972992

does not contain the digit 3, while each number 2k, for k = 154, 155, . . . , 3000,

contains it (see the number 71);

• the value of 1! + 2! + 3! + 4! + 5! .

154

• the rank of the prime number which appears the most often as the 11th prime

factor of an integer : p154 = 887 (see the number 199);

• the 11th number n such that n! + 1 is prime (see the number 116).

155

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

229

+ 1,

a number whose complete factorization was obtained in 1990 (see the number

70 525 124 609).

156

• the second number n 1 which divides σ8(n) (see the number 84).