Those Fascinating Numbers 95
588
the smallest number n 2 such that
σ(n) + φ(n)
γ(n)2
is an integer: the sequence
of numbers satisfying this property begins as follows: 1, 2, 588, 864, 2430,
7776, 27000, 55296, 69984, 82134, 215622, 432000, 497664, 629856, 675000,
862488,. . .
99;
the third number n such that φ(n) + σ(n) = 3n (see the number 312).
590
the rank of the prime number which appears the most often as the
14th
prime
factor of an integer: p590 = 4297 (see the number 199).
595
the third number n such that φ(n)σ(n) is a fourth power: φ(595)σ(595) =
244
(see the number 170).
596
the third composite number n such that σ(n + 6) = σ(n) + 6 (see the number
104).
598 (= 2 · 13 · 23)
the fifth number n 9 such that n =
∑r
i=1
di, i where d1, . . . , dr stand for the
digits of n: here 598 = 51 + 92 + 83 (see the number 175);
the smallest square-free composite number n such that p|n =⇒ p + 2|n + 2:
the sequence of numbers satisfying this property begins as follows: 598, 3913,
11590, 32578, 91078, 95170, 154843, 179998, 301273, 317623, . . . (see the num-
ber 399).
602
the smallest number n such that Ω(n) = Ω(n + 1) = Ω(n + 2) = Ω(n + 3): here
this common value is 3; if nk stands for the smallest number n such that Ω(n) =
Ω(n+1) = . . . = Ω(n+k−1), then n2 = 14, n3 = 33, n4 = n5 = 602, n6 = 2 522,
n7 = 211 673, n8 = n9 = 3 405 122, n10 = 49 799 889, n11 = 202 536 181 and
n12 = 3 195 380 879 (for the analogue question with the function ω(n), see the
number 44 360);
the number of prime numbers revealed by the 1001 first values of the polynomial
2n2 1000n 2609 introduced by H.S. Williams (R.K. Guy [101], A17 (with an
error: one should read 1001 instead of 1000), and R.A. Mollin [140]).
99It
is easy to show that there exist infinitely many numbers n such that
σ(n) + φ(n)
γ(n)2
is an integer,
namely by considering the numbers n = 32 · 32r+1, r = 1, 2, 3, . . . .
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