Those Fascinating Numbers 159

3 913

• the second square-free composite number n such that p|n =⇒ p + 2|n + 2 (see

the numbers 598 and 399).

3 994

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

19th

prime

factor of an integer : p3994 = 37 717 (see the number 199).

4 030

• the third bizarre number (see the number 70).

4 093

• the

13th

prime number pk such that p1p2 . . . pk − 1 is prime (see the number

317).

4 095

• the largest triangular number of the form 2r − 1: here 4 095 = 212 − 1 =

90·91

2

;

there exist only four numbers satisfying this property, the other three being

21 − 1, 22 − 1 =

2·3

2

and 24 − 1 =

5·6

2

;

• the smallest number n such that 8! divides 1+2+ . . . + n (see the number 224);

• the sixth odd abundant number (see the number 945).

4 104

• the second number which can be written as the sum of two cubes in two distinct

ways: 4 104 =

23

+

163

=

93

+

153

(see the number 1 729).

4 140

• the eighth Bell number (see the number 52).

4 150

• the smallest number n 1 which can be written as the sum of the fifth powers of

its digits: 4 150 = 45 + 15 + 55 + 05; the only numbers satisfying this property

are 1, 4 150, 4 151, 54 748, 92 727, 93 084 and 194 979; if nk stands for the

smallest number n 1 which can be written as the sum of the k-th power of

its digits, then n3 = 153, n4 = 1 634, n5 = 4 150, n6 = 548 834, n7 = 1 741 725,

n8 = 24 678 050, n9 = 146 511 208 and n10 = 4 679 307 774.