A261385 Least positive integer k such that (prime(prime(k))-1)*(prime(prime(k*n))-1) = prime(p)-1 for some prime p.

1, 3, 221, 15, 13, 137, 63, 103, 44, 2, 31, 3, 45, 3, 4, 104, 38, 237, 61, 19, 56, 183, 22, 11, 15, 374, 9, 5, 42, 97, 2, 47, 4, 19, 23, 399, 3, 103, 29, 10, 2, 109, 51, 1, 52, 80, 23, 64, 76, 2, 218, 3, 7, 98, 4, 145, 10, 12, 213, 87, 36, 181, 28, 169, 71, 25, 72, 71, 54, 50

Offset: 1

Author

Zhi-Wei Sun, Aug 17 2015

Keywords

• nonn

Comments

Conjecture: Let d be any nonzero integer. Then each positive rational number r can be written as m/n, where m and n are positive integers with (prime(prime(m))+d)*(prime(prime(n))+d) = prime(p)+d for some prime p.

This conjecture implies that for any nonzero integer d the equation x*y = z with x,y,z in the set {prime(p)+d: p is prime} has infinitely many solutions.

Examples

			a(3) = 221 since (prime(prime(221))-1)*(prime(prime(221*3))-1) = (prime(1381)-1)*(prime(4957)-1) = 11446*48130 = 550895980 = prime(28890079)-1 with 28890079 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
• Zhi-Wei Sun, Checking the conjecture for d = -1 and r = a/b (a,b = 1..60)
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A257938, A261353.

Programs

Mathematica

f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]+1],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]