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%I #16 Aug 01 2013 09:39:44
%S 2,3,11,17,23,29,41,43,59,61,71,73,79,97,101,103,107,131,137,149,151,
%T 163,167,179,191,193,223,227,229,239,251,257,269,271,277,281,311,331,
%U 347,349,353,359,367,373,383,397,419,421,431,433,439,457,461,463,479,491
%N Primes p with the property: 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 (p, 3p_(m+1)) contains a prime.
%C For k>1 (not necessarily integer), we call a Ramanujan k-prime R_n^(k) the prime a_k(n) which is the smallest number such that if x >= a_k(n), then pi(x)- pi(x/k) >= n. Note that, the sequence of all primes corresponds to the case of "k=oo". These numbers possess the following properties: R_n^(k)~p_((k/(k-1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan k-primes not exceeding x, then A_k(x)~(1-1/k)\pi(x) as x tends to the infinity; let p be a Ramanujan k-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 (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist non-Ramanujan k-primes, which possess the latter property. For example, for k=2, the smallest such prime is 109. Problem. For every k>2 to estimate the smallest non-Ramanujan k-prime which possesses the latter property. [From Vladimir Shevelev, Sep 01 2009]
%C All Ramanujan 3-primes are in the sequence.
%H Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%e If p=61, the p/3 is in the interval (19, 23); we see that the interval (61,69) contains a prime (67).
%t nn=1000; t=Table[0, {nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s<nn, t[[s+1]]=k], {k, Prime[3*nn]}]; t=t+1
%Y Cf. A104272, A164368, A164288.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Sep 01 2009
%E Extended and edited by _T. D. Noe_, Nov 22 2010
%E Comments edited by _Jonathan Sondow_, Aug 27 2011
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