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 = · 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.
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