86 Jean-Marie De Koninck

420

• the largest number n such that 4τ (n) = φ(n); the only solutions of this equation

are n = 34, 45, 52, 102, 140, 156, 252, 360 and 420.

427

• the

15th

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

429

• the seventh Catalan number (see the number 14).

430

• the smallest solution of σ2(n) = σ2(n + 4); the sequence of numbers satisfy-

ing this equation begins as follows: 430, 2 158, 1 895 038, 2 724 478, 4 460 542,

29 879 998, 39 440 014, 65 018 878, 91 163 518, 91 682 494, 98 873 854, . . .

89.

431

• the smallest prime factor of the Mersenne number 243 −1 (Landry, 1869), whose

complete factorization is given by

243

− 1 = 8 796093 022 207 = 431 · 9719 · 2099863.

432

• the smallest number n such that n! is not a Niven number; indeed, the sum of

the digits of 432! is equal to 3897 = 9 · 433, a quantity which does not divide

432!, while one can check that n! is a Niven number for each number n ≤ 431.

433

• the 11th Markoff number, namely the 11th solution z of equation x2 + y2 + z2 =

3xyz, where x ≤ y ≤ z: the sequence of Markoff numbers begins as follows: 1,

2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, . . . (R.K. Guy [101], D12).

434

• the smallest composite number n such that σ(n + 2) − σ(n) = 2: the only

composite numbers n

109

satisfying this equation are 434, 8 575 and 8 825

(see also the number 305 635 357).

89One can use the solutions of σ2(n) = σ2(n + 2) to generate the solutions of σ2(n) = σ2(n + 4);

indeed, by considering the even numbers n = 2m, with m odd, it is clear that equation σ2(2m) =

σ2(2m + 4) is equivalent to equation σ2(m) = σ2(m + 2).