A387196 - OEIS

%I #21 Aug 29 2025 19:25:04

%S 13,17,19,20,21,25,36,37,45,49,55,91,105,127,169,181,187,247,307,361,

%T 391,429,541,577,667,811,937,961,969,1147,1297,1567,1591,1801,1849,

%U 1927

%N Integers k such that 1/k = (1/p - 1/q)*(1/r - 1/s) for distinct primes p < q and r < s.

%C For any prime p, an exhaustive search with primes up to p finds all terms t in the sequence that satisfy t < next_prime(p).

%C If p and p+d are primes with d in {2,6}, then 6*p*(p+d)/d is in the sequence.

%C If p and p+2 are primes, then (p+2)^2 is in the sequence.

%C If p is a prime such that p = (b+1)*(c-1)+1 for some primes b and c with c-b also prime, then p is in the sequence.

%e 1/13 = (1/2 - 1/5)*(1/3 - 1/13),

%e 1/17 = (1/3 - 1/5)*(1/2 - 1/17),

%e 1/20 = (1/2 - 1/3)*(1/2 - 1/5),

%e 1/36 = (1/2 - 1/3)*(1/2 - 1/3),

%e 1/45 = (1/2 - 1/3)*(1/3 - 1/5).

%K nonn,hard,more

%O 1,1

%A _Yuto Tsujino_, Aug 21 2025

%E a(30)-a(36) from _Hugo Pfoertner_, Aug 23 2025