cp's OEIS Frontend

A164958 Primes p with the property that if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (3p_m, p) contains a prime.

%I A164958 #7 Jul 22 2025 07:19:06
%S A164958 2,3,5,13,19,29,31,43,47,61,67,73,79,83,101,103,107,109,137,139,151,
%T A164958 157,167,173,181,193,197,199,229,233,241,257,263,271,277,281,283,313,
%U A164958 317,349,353,359,367,373,379,389,401,409,431,433,439,443,461,463,467,487,499
%N A164958 Primes p with the property that if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (3p_m, p) contains a prime.
%C A164958 For k>1 (not necessarily integer), we call a Labos k-prime L_n^(k) the prime a_k(n) which is the smallest number such that pi(a_k(n)) - pi(a_k(n)/k)= n. Note that, the sequence of all primes corresponds to the case of "k=oo". Let p be a k-Labos prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the n-th prime. Then the interval (k*p_(m), p) contains a prime. Conjecture. For every k>1 there exist non-k-Labos primes, which possess the latter property. For example, for k=2, the smallest such prime is 131. Problem. For every k>1 to estimate the smallest non-k-Labos prime, which possess the latter property. [From _Vladimir Shevelev_, Sep 02 2009]
%C A164958 All 3-Labos primes are in this sequence.
%e A164958 If p=61, the p/3 is in the interval (19, 23); we see that the interval (57, 61) contains a prime (59). Thus 61 is in the sequence.
%t A164958 nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]
%Y A164958 Cf. A104272, A080359, A164952, A164368, A164288, A164294
%K A164958 nonn
%O A164958 1,1
%A A164958 _Vladimir Shevelev_, Sep 02 2009
%E A164958 Extended by _T. D. Noe_, Nov 23 2010