Those Fascinating Numbers 221

50 811

• the smallest number n which allows the sum

m≤n

1

φ(m)

to exceed 21 (see the

number 177).

51 169

• the

41rst

(and largest known) Lucas prime number (see the number 613).

51 301 (=

292

· 61)

• the smallest (and possibly the only) 19-hyperperfect number.

51 984

• the

11th

powerful number which can be written as the sum of two co-prime

3-powerful numbers = 1: 51 984 = 1 331+50 653, that is

24

·

32

·

192

=

113 +373

(see the number 841).

52 020

• the sixth number n such that σ(n) and σ2(n) have the same prime factors,

namely the primes 2, 3, 7, 13 and 307 (see the number 180).

52 488

• the

100th

3-powerful number (see the number 216).

54 178

• the smallest even number n such that σI (n) = σI (n + 2); the sequence of

numbers satisfying this property begins as follows: 54 178, 230 578, 467 458,

5 058 180, 5 761 978, 28 119 418, 34 385 818, 35 080 502, 37 387 980, . . .

54 748

• the fourth number which can be written as the sum of the fifth powers of its

digits: 54 748 = 55 + 45 + 75 + 45 + 85 (see the number 4 150).

55 296

• the sixth number n 2 such that

σ(n) + φ(n)

γ(n)2

is an integer (see the number

588).