Those Fascinating Numbers 111

882

• the

15th

number n such that n ·

2n

− 1 is prime (see the number 115).

887

• the prime number which appears the most often as the

11th

prime factor of an

integer (see the number 199);

• the smallest prime number p such that p + 20 is prime and such that each

number between p and p + 20 is composite (see the number 139);

• the smallest prime factor of the Mersenne number 2443 − 1, whose complete

factorization is given by

2443

− 1 = 887 · 207818990653657 · P117;

• the second number which does not produce a palindrome by the 196-algorithm

(see the number 196).

891

• the second odd number n 1 such that γ(n)|σ(n) (see the number 135).

906

• the smallest number n such that inequality log g(m) ≥

√

m log m holds for

all m ≥ n: here g(m) = max

σ∈Sm

(order of σ), where Sm stands for the group of

permutations of m (J.P. Massias [131]).

907

• the smallest prime number which is preceded by 19 consecutive composite num-

bers; indeed, there is no prime number between 887 and 907.

911

• the first component of the fifth pair of prime numbers {p, q} such that

pq−1

≡ 1 (mod

q2)

and

qp−1

≡ 1 (mod

p2);

here {p, q} = {911, 318917} (see the number 2 903).

915

• the smallest odd number which can be written as the sum of some

powers117

of its prime factors: here 915 = 3 · 5 · 61 =

36

+

53

+

611.

117In

2005, J.M. De Koninck & F. Luca [54] studied the size of the set {n ≤ x : ω(n) ≥ 2 and n = ∑

p|n

pαp }, where each exponent αp can vary with the prime divisor p.