86 Jean-Marie De Koninck
420
the largest number n such that (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).
Previous Page Next Page