Those Fascinating Numbers 19

54

• the number of digits in the decimal expansion of the tenth perfect number

288(289

− 1).

55

• the only solution n 108 of φ(n)σ(n) = φ(n + 1)σ(n + 1) = φ(n + 2)σ(n + 2)

(see R.K. Guy [102]; see also the number 136);

• the largest Fibonacci number with only one distinct digit, a result due to

F. Luca; the sequence of Fibonacci numbers (Fn)n≥1 is defined as follows:

F1 = F2 = 1, Fn = Fn−1 + Fn−2 for each n ≥ 3;

• the largest Fibonacci number which is the concatenation of two other Fibonacci

numbers (5 and 5), a result due to F. Luca & W.D. Banks [16];

• the fourth and largest Fibonacci number which is triangular: the others are 1,

3 and 21 (see L. Ming [137]).

56

• the sixth tetrahedral number (see the number 10): the sequence of tetrahedral

numbers begins as follows: 1, 4, 10, 20, 35, 56, 84, 120, . . . ;

• the largest number n for which there exists a number k ≤ n/2 such that if we

write the binomial coeﬃcient

(n)

k

as a product uv, where

u =

p|(n)

k

pk

p et v =

p|(n)k

p≥k

p,

then u v: the only other pairs (n, k) satisfying this property are (8, 3), (9, 4),

(10, 5), (12, 5), (21, 7), (21, 8), (30, 7), (33, 13), (33, 14), (36, 13), (36, 17) and

(56, 13) (E.F. Ecklund, R.B. Eggleton, P. Erd˝ os & J.L. Selfridge [74]).

57

• the smallest solution of σ(n) = σ(n + 22); the sequence of numbers satisfying

this equation begins as follows: 57, 85, 213, 224, 354, 476, 568, 594, 812, 1218,

1235, 1316, 1484, 2103, 2470, 2492, 2643, 2840, 2996, 3836, 3978, 4026, 4544,

4810, 4844, 5012, 6125, 6356, 6524, 7364, 7532, 7648, 8876, 9272, 9328, . . . ;

• the smallest number n such that

∑

m≤n

φ(m) is a multiple of 1 000 (here the

sum is equal to 1 000); the sequence of numbers satisfying this property begins

as follows: 57, 140, 303, 358, 862, . . .