204 Jean-Marie De Koninck

23 526

• the smallest number n such that π(n) = n/9 (see the number 330).

23 569

• the seventh number whose square can be written as the sum of three fourth

powers: 23

5692

=

844

+

1054

+

1404

(see the number 481).

23 760

• the smallest number n such that φ(n) + σ(n) = 4n; the sequence of numbers

satisfying this property begins as follows: 23 760, 59 400, 153 720, 4 563 000,

45 326 160, 113 315 400, 402 831 360, 731 601 000, 803 685 120, 865 950 624,

919 501 200, 1 178 491 680, 3 504 597 120, 3 786 686 400 and 6 429 564 000 (the

largest 1010); F. Luca & J. Sandor [129] obtained interesting results on com-

posite numbers n for which the quotient

φ(n) + σ(n)

n

is an integer159 (see also

R.K. Guy [102] as well as the number 312).

23 762

• the fourth number n divisible by a square 1 and such that δ(n+1)−δ(n) = 1

(see the number 49).

23 801

• the

17th

prime number pk such that p1p2 . . . pk + 1 is prime (see the number

379).

23 805

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

namely the primes 2, 3, 7, 13 and 79 (see the number 180).

24 029

• the 18th prime number pk such that p1p2 . . . pk + 1 is prime (see the number

379).

159If

A stands for the set of numbers n for which

φ(n) + σ(n)

n

is an integer, then Luca and Sandor

obtained various results concerning the nature of the set A; in particular, they proved that for each

k ≥ 2, the set A contains only a finite number of numbers n with Ω(n) ≤ k, and that if n ∈ A and

ω(n) = 3, then either n = 2α · 3 · p with p = 2α−2 · 7 − 1 prime, or else n ∈ {560, 588, 1400}. They

also established an upper bound for the number of elements of A not exceeding x. C.A. Nicol’s

conjecture which claims that all elements of A are even has not yet been proved.