A392682 - OEIS

A392682

Primes p(k) > 3 such that, for every integer 1 < m < k (except m = k - 2 if p(k) > 5), there exists a set of m distinct primes < p(k) the sum of whose reciprocals forms a fraction that has a numerator divisible by p(k).

%I #16 Feb 01 2026 20:49:17

%S 5,19,109

%N Primes p(k) > 3 such that, for every integer 1 < m < k (except m = k - 2 if p(k) > 5), there exists a set of m distinct primes < p(k) the sum of whose reciprocals forms a fraction that has a numerator divisible by p(k).

%C Conjecture: this is the intersection of A006512 and A293457.

%C a(4) > 10^8.

%H Arkadiusz Wesolowski, <a href="/A392682/a392682.txt">Examples of sets for a(3) = 109</a>

%e Examples of sets for a(2) = 19:

%e {2, 17},

%e {11, 13, 17},

%e {2, 3, 5, 7},

%e {3, 5, 7, 11, 13},

%e {2, 3, 5, 7, 11, 13, 17}.

%Y Cf. A000720, A006512, A293457.

%K nonn,bref,hard,more

%O 1,1

%A _Arkadiusz Wesolowski_, Jan 19 2026