306 Jean-Marie De Koninck

25 417 732

• the smallest number 1 which is equal to the sum of the squares of the

factorials of its digits in base 8: here 25 417 732 = [1, 4, 0, 7, 5, 4, 0, 0, 4]8 =

1!2 +4!2 +0!2 +7!2 +5!2 +4!2 +0!2 +0!2 +4!2 (see the numbers 145 and 40 585).

25 430 981

• the second number 1 which is equal to the sum of the squares of the factorials

of its digits in base 8: here 25 430 981 = [1, 4, 1, 0, 0, 5, 7, 0, 5]8 = 1!2 +4!2 +1!2 +

0!2

+

0!2

+

5!2

+

7!2

+

0!2

+

5!2

(see the number 25 417 732).

25 457 760

• the eighth number n such that β(n)|β(n+1) and β(n+1)|β(n+2): here 164|984

and 984|1157184 (see the number 225 504).

25 658 441

• the first component p of the third 8-tuple (p, p +2, p +6, p +8, p +12, p +18, p +

20, p + 26) made up entirely of prime numbers: the smallest such 8-tuple is

(11, 13, 17, 19, 23, 29, 31, 37), while the second is the one whose first component

is 15 760 091.

25 741 470

• the fourth solution of σ(n) = σ(n + 15) (see the number 26).

25 964 951

• the exponent of the

42nd

Mersenne prime

225 964 951

− 1 (a 7 816 230 digit num-

ber) discovered by Martin Nowak (an eye surgeon) on February 18, 2005, using

the programme developed by G. Woltman (see the number 1 398 269).

26 888 999

• the smallest number of persistence 9 (see the number 679).

26 890 623

• the first of the smallest three consecutive numbers each divisible by a sixth

power: 26 890 623 = 36 ·36887, 26 890 624 = 27 ·19·11057, 26 890 625 = 56 ·1721

(see the number 1 375).