Those Fascinating Numbers 127

1 386

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 14 (see the

number 177).

1 393

• the smallest number which can be written both as the sum of three fourth

powers and as the sum of four fourth powers: 1393 =

24

+

34

+

64

=

44

+

44

+

44

+

54.

1 395

• the second vampire number; in this case, 1 395 = 93·15 (see the number 1 260).

1 399

• the smallest prime factor of the Mersenne number

2233

− 1, whose complete

factorization is given by

2233

− 1 = 1399 · 135607 · 622577

·116868129879077600270344856324766260085066532853492178431.

1 400

• the fourth number n such that φ(n) + σ(n) = 3n (see the number 312);

• the second multiple of 100 such that the next 100 numbers contain exactly 17

prime numbers, namely 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453,

1459, 1471, 1481, 1483, 1487, 1489, 1493 and 1499 (see the number 400).

1 415

• the fourth composite number n such that σ(n + 6) = σ(n) + 6 (see the number

104).

1 428

• the smallest 4-composite number n such that n+2 is also a 4-composite number:

we say that a number n is k-composite if ω(n) = k: here 1428 =

22

· 3 · 7 · 17

and 1430 = 2 · 5 · 11 · 13; denoting by nk the smallest k-composite number n

such that n + 2 is also k-composite, we have the following table:

k 1 2 3 4 5 6 7 8

nk 2 10 130 1 428 77 140 1 456 728 117 048 930 10 326 137 820