214 Jean-Marie De Koninck

39 999

• the smallest solution of τ (n + 1) − τ (n) = 27 (see the number 399).

40 001

• the third composite number of the form k · 10k + 1 (see the number 201).

40 311

• the eighth number n such that n, n + 1, n + 2, n + 3 and n + 4 have the same

number of divisors, namely eight (see the number 11 605).

40 320

• the value of 8! .

40 369

• the second composite number n such that 2n−4 ≡ 1 (mod n); the only com-

posite numbers n 108 satisfying this property are 4, 40 369, 673 663, 990 409,

1 697 609, 2 073 127, 6 462 649, 7 527 199, 7 559 479, 14 421 169, 21 484 129,

37 825 753 and 57 233 047.

40 391

• the sixth solution w of the aligned houses problem (see the number 35).

40 465

• the smallest number which is equal to the sum of the factorials of its digits in

base 17: here 40 465 = [8, 4, 0, 5]17 = 8! + 4! + 0! + 5! (see the numbers 145 and

40 472).

40 472

• the largest number which is equal to the sum of the factorials of its digits in

base 11: here 40 472 = [2, 8, 4, 5, 3]11 = 2! + 8! + 4! + 5! + 3!; the only numbers

with this property are 1, 2, 26, 48 and 40 472 (see also the number 145).

40 487

• the third odd prime number p such that

5p−1

≡ 1 (mod

p2)

(see the number

20 771);

• the smallest prime number p whose smallest primitive root modulo p (namely

here 5) is not a primitive root modulo

p2

(a problem stated by Neville Rob-

bins at the 1998 West Coast Number Theory Conference and solved by many

participants).