Those Fascinating Numbers 197

18 287

• the smallest prime factor of the Mersenne number

2223

− 1, whose complete

factorization is given by

2223

− 1 = 18287 · 196687 · 1466449 · 2916841

·1469495262398780123809 · 596242599987116128415063;

it is the smallest Mersenne number with exactly six prime factors (see the num-

ber 223 for the list of the smallest numbers which require a given number of

distinct primes).

18 315

• the 21rst solution of φ(n) = φ(n + 1) (see the number 15).

18 444

• the smallest number n such that ω(n) + ω(n + 1) + ω(n + 2) + ω(n + 3) = 15:

here 18 444 = 22 · 3 · 29 · 53, 18 445 = 5 · 7 · 17 · 31, 18 446 = 2 · 23 · 401 and

18 447 = 3 · 11 · 13 · 43 (see the number 987).

18 486

• the third composite number n (and the largest 109) which divides the ex-

pression σ(n) + φ(n) + γ(n) (see the number 198).

18 490

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

(see the number 49).

18 496

• the fifth number n 1 such that n · 2n + 1 is prime (see the number 141).

18 523

• the 16th prime number pk such that p1p2 . . . pk + 1 is prime (see the number

379).

18 600

• the eighth number which is not perfect or multi-perfect but whose harmonic

mean is an integer (see the number 140).