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].
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