Those Fascinating Numbers 355

38 358 837 677

• the smallest known prime number p for which inequality

π2(p)

ep

log p

π

p

e

does not hold; Ramanujan proved that this inequality holds for p suﬃciently

large (see B.C. Berndt [21], as well as the number 2 418).

43 861 478 400 (=

210

·

33

·

52

· 23 · 31 · 89)

• the eighth 4-perfect number (see the number 30 240).

44 496 177 152

• the smallest number with an index of composition 2 which can be written as

the sum of two co-prime numbers each with an index of composition ≥ 7: we

have

44 496 177 152 =

211

·

74

· 9049 =

198

+

317,

where

λ(211

·

74

· 9049) ≈ 2.08679,

λ(198)

= 8,

λ(317)

= 7

(see the number 607 323 321).

51 001 180 160

(=214

· 5 · 7 · 19 · 31 · 151)

• the sixth tri-perfect number and the largest one known (see the number 120).

52 523 350 144 (=

347)

• the fourth number n 1 whose sum of digits is equal to

7

√

n (see the number

612 220 032).

55 420 693 056

• the seventh number which is both triangular and a perfect square: 55 420 693 056 =

332928(332928+1)

2

= 2354162 (see the number 36).

61 917 364 224

• the smallest fifth power which can be written as the sum of four fifth powers:

61 917 364 224 =

1445

=

275

+

845

+

1105

+

1335

(Lander & Parkin [122]).