A261385 - OEIS

%I #10 Nov 16 2024 02:02:28

%S 1,3,221,15,13,137,63,103,44,2,31,3,45,3,4,104,38,237,61,19,56,183,22,

%T 11,15,374,9,5,42,97,2,47,4,19,23,399,3,103,29,10,2,109,51,1,52,80,23,

%U 64,76,2,218,3,7,98,4,145,10,12,213,87,36,181,28,169,71,25,72,71,54,50

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

%C 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.

%C 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.

%H Zhi-Wei Sun, <a href="/A261385/b261385.txt">Table of n, a(n) for n = 1..1000</a>

%H Zhi-Wei Sun, <a href="/A261385/a261385.txt">Checking the conjecture for d = -1 and r = a/b (a,b = 1..60)</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e 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.

%t f[n_]:=Prime[Prime[n]]-1

%t PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]

%t 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}]

%Y Cf. A000040, A257938, A261353.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Aug 17 2015