Those Fascinating Numbers 77

348

• the smallest number n such that n, n +2, n +4, n +6 are abundant (we say that

a number n is abundant if σ(n) 2n): the sequence (nk)k≥1 of the smallest

numbers n = nk such that n, n + 2, . . . , n + 2k are abundant begins as follows:

18, 100, 348, 2 988, 801 340, . . . ;

• the smallest solution of τ (n + 12) = τ (n) + 12; the sequence of numbers satis-

fying this equation begins as follows: 348, 492, 758, 788, 852, 898, 933, 1164,

1212, 1236, . . .

353

• the smallest number whose fourth power can be written as the sum of four

fourth powers: 3534 = 304 + 1204 + 2724 + 3154 (R.Norrie, 1911): see R.K. Guy

[101], D1;

• the third prime number of the form

(x4

+

y4)/2:

here 353 =

(54

+

34)/2

(see

the number 41).

357

• the second square-free composite number n for which p|n =⇒ p + 3|n + 3 (see

the number 165).

359

• the smallest prime factor of the Mersenne number

2179

− 1, whose complete

factorization is given by

2179

− 1 = 359 · 1433 · 1489459109360039866456940197095433721664951999121.

360 (=

23

·

32

· 5)

• the smallest number m such that equation σ(x) = m has exactly nine solutions,

namely 120, 174, 184, 190, 267, 295, 319, 323 and 359;

• the smallest solution of

σ(n)

n

=

13

4

; the sequence of numbers satisfying this

equation begins as follows: 360, 2 016, 1 571 328, 4 428 914 688,. . . ;

• the 13th highly composite number (see the number 180).

361

• the smallest Smith number (see the number 22) which is a perfect square:

361 = 192 and 3 + 6 + 1 = 10 = 1 + 9;

• the number at which the function (ψ(x) − θ(x))/

√

x reaches its maximal value,

where θ(x) =

p≤x

log p and ψ(x) =

pm

≤x

m≥1

log p (J.B. Rosser & L. Schoenfeld [178]).