Those Fascinating Numbers 291

5 761 455

• the number of prime numbers

108.

5 761 978

• the fifth even number n such that σI (n) = σI (n + 2) (see the number 54 178).

5 909 560

• the smallest number n such that P (n) ≤

5

√

n and P (n + 1) ≤

5

√

n + 1: here

P (5909560) = P

(28

·

35

· 5 · 19) = 19 22.61 ≈ 5

√

5909560 and P (5909561) =

P

(112

·

132

·

172)

= 17 22.61 ≈ 5

√

5909561; the sequence of numbers satis-

fying this property begins as follows: 5909560, 11859210, 71843750, 76271624,

80061344, 96059600, 119094299, 132663167, 133919999, 177182720,

181037024,. . . (see the number 2

400)189.

5 978 882

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

powers in four distinct ways:

5 978 882 =

34

+

404

+

434

=

84

+

374

+

454

=

154

+

324

+

474

=

234

+

254

+

484

(see the number 6 578).

6 082 047

• the smallest number n such that n and n+1 each have 11 prime factors counting

their multiplicity: 6 082 047 =

310

· 103 and 6 082 048 =

29

· 7 · 1697 (see the

number 135).

6 197 024

• the smallest number n such that φ(6n+1) φ(6n+2) (see D.J. Newman [149]).

6 303 734

• the fourth number n such that Eσ(n) := σ(n + 1) − σ(n) satisfies Eσ(n + 1) =

Eσ(n): here the common value of Eσ is 1 429 056, since σ(6303734) = 9656928,

σ(6303735) = 11085984 and σ(6303736) = 12515040 (see the number 693).

6 436 343

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

6 436 343 =

235

=

15

+

75

+

85

+

145

+

155

+

185

+

205.

189This sequence is infinite: see the footnote tied to the number 2 400.