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, . . . .