Those Fascinating Numbers 81

• the 14th number n such that n! − 1 is prime (see the number 166).

381

• the smallest number larger than 1 and whose sum of divisors is a ninth power:

σ(381) = 29.

382

• the smallest solution of σ(n) = σ(n + 3); the sequence of numbers satisfying

this equation begins as follows: 382, 8922, 11935, 31815, 32442, 61982, 123795,

145915, 186615, 271215, . . .

383

• the smallest prime factor of the Mersenne number 2191 − 1, whose complete

factorization is given by

2191

− 1 = 383 · 7068569257 · 39940132241

·332584516519201 · 87274497124602996457;

• the smallest prime number which can be written as the sum of a prime number p

and of the number obtained by reversing the digits of p: indeed, 241+142 = 383

(an observation due to Shyam Sunder Gupta).

384

• the 11th number n such that n · 2n − 1 is prime (see the number 115);

• the eighth Granville number (see the number 126).

386

• the smallest number n 1 such that n|σ144(n).

389

• the smallest prime number whose sum of digits is 20; for each number k ≥ 2,

not a multiple of 3, let ρ(k) stand for the smallest prime number whose sum of

digits is k, here are the values of ρ(k) for k = 2, 4, 5, 7, 8, . . . , 83: 85

85It

is easy to prove that ρ(k) ≥ (a +

1)10b

− 1, where b = [k/9] and a = k − 9b. However, note

that it is not clear that ρ(k) is well defined for all integers k 1 which are not a multiple of 3. On

the other hand, if β(k) stands for the number of prime numbers whose sum of digits is equal to k,

it is easy to see that there exist infinitely many positive integers k such that β(k) ≥ 1. It is also

possible to prove that there exist infinitely many positive integers k such that ρ(k) ≤

102k.

Finally,

several questions regarding the ρ function can be raised; here are some of them:

– does there exist infinitely many k’s such that β(k) = +∞ ?

– is it true that β(2) ≥ 3 ?

– is it true that limk→∞ ρ(k)/((a +

1)10b

− 1) = 1, where b = [k/9] and a = k − 9b ?

for more on this problem, see J.M. De Koninck [46].