248 Jean-Marie De Koninck

213 444 (= 3622)

• the second perfect square whose last three digits are 444 (see the number 1 444).

214 195

• the third number n such that σ(n), σ(n + 1), σ(n + 2), σ(n + 3) and σ(n + 4)

have the same prime factors, namely here 2, 3, 5, 7 and 17 (see the numbers

3 777 and 20 154).

214 273 (= 472 · 97)

• the smallest 31-hyperperfect number (see the number 21).

214 369 (=

4632)

• the smallest powerful number n such that n + 6 is also powerful: here n + 6 =

54 · 73; the sequence of numbers satisfying this property begins as follows:

214369, 744769, 715819215721, . . . 175 (see the number 463).

215 326

• the second even number n such that

2n

≡ 2 (mod n) (see the number 161 038).

215 622

• the ninth number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).

216 091

• the exponent of the

31rst

Mersenne prime

2216 091

− 1 (Slowinski, 1985).

217 070

• the smallest number n such that n, n + 1, n + 2, n + 3, n + 4, n + 5 and n + 6 are

all divisible by a square 1: here 217 070 = 2 · 5 ·

72

· 443, 217 071 =

32

· 89 · 271,

217 072 =

24

· 13567, 217 073 = 17 ·

1132,

217 074 = 2 · 3 ·

112

· 13 · 23, 217 075 =

52

· 19 · 457 and 217 076 =

22

· 54269 (see the number 242).

175This sequence is infinite, since in 1982, W.L. Daniel [41] proved that each positive integer can

be written as the difference of two powerful numbers in infinitely many ways. Moreover, a few

years later, R.A. Mollin and P.G. Walsh [141] proved that each integer m 0 has infinitely many

representations as m = P − Q, where P and Q are powerful numbers neither of them being a perfect

square.