A387196 Integers k such that 1/k = (1/p - 1/q)*(1/r - 1/s) for distinct primes p < q and r < s.

13, 17, 19, 20, 21, 25, 36, 37, 45, 49, 55, 91, 105, 127, 169, 181, 187, 247, 307, 361, 391, 429, 541, 577, 667, 811, 937, 961, 969, 1147, 1297, 1567, 1591, 1801, 1849, 1927

Offset: 1

Author

Yuto Tsujino, Aug 21 2025

Keywords

• nonn
• hard
• more
• new

Comments

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).

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

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

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.

Examples

			1/13 = (1/2 - 1/5)*(1/3 - 1/13),
1/17 = (1/3 - 1/5)*(1/2 - 1/17),
1/20 = (1/2 - 1/3)*(1/2 - 1/5),
1/36 = (1/2 - 1/3)*(1/2 - 1/3),
1/45 = (1/2 - 1/3)*(1/3 - 1/5).
		

Extensions

a(30)-a(36) from Hugo Pfoertner, Aug 23 2025