Those Fascinating Numbers 331

479 045 521

• the smallest number n such that φ(n) = 12! (see the number 779).

492 101 632 (= 213 · 11 · 43 · 127)

• the fourth solution of

σ(n)

n

=

9

4

(see the number 40).

519 326 767

• the only number 2 which is equal to the sum of the factorials of its digits

in base 13: here 519 326 767 = [8, 3, 7, 9, 0, 12, 5, 11]13 = 8! + 3! + 7! + 9! + 0! +

12! + 5! + 11! (see also the numbers 145 and 40 472).

528 491 311

• the integer part of

ee3

, which represents the “normal” size of the third prime

factor of an integer (see the number 1 618).

534 494 836

• the third number n 1 which is equal to the sum of the ninth powers of its

digits (see the number 146 511 208).

537 346 048

• the

21rst

dihedral perfect number (see the number 130).

565 062 156

• the smallest number n such that P (n) P (n + 1) . . . P (n + 11): here

1553 3541 8377 19993 23623 31321 189491 752413

3445501 12556937 14870057 33238951 (see the number 46 189).

592 192 224

• the 12th powerful number n such that n +1 is also powerful: here 592 192 224 =

25 · 32 · 132 · 233 and 592 192 225 = 52 · 312 · 1572 (see the number 288).