

A164952


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 kth prime, then the interval (p, 3p_(m+1)) contains a prime.


8



2, 3, 11, 17, 23, 29, 41, 43, 59, 61, 71, 73, 79, 97, 101, 103, 107, 131, 137, 149, 151, 163, 167, 179, 191, 193, 223, 227, 229, 239, 251, 257, 269, 271, 277, 281, 311, 331, 347, 349, 353, 359, 367, 373, 383, 397, 419, 421, 431, 433, 439, 457, 461, 463, 479, 491
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OFFSET

1,1


COMMENTS

For k>1 (not necessarily integer), we call a Ramanujan kprime 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/(k1))n) as n tends to the infinity; if A_k(x) is the counting function of the Ramanujan kprimes not exceeding x, then A_k(x)~(11/k)\pi(x) as x tends to the infinity; let p be a Ramanujan kprime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the nth prime. Then the interval (p, k*p_(m+1)) contains a prime. Conjecture. For every k>=2 there exist nonRamanujan kprimes, 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 nonRamanujan kprime which possesses the latter property. [From Vladimir Shevelev, Sep 01 2009]
All Ramanujan 3primes are in the sequence.


LINKS

Table of n, a(n) for n=1..56.
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785


EXAMPLE

If p=61, the p/3 is in the interval (19, 23); we see that the interval (61,69) contains a prime (67).


MATHEMATICA

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


CROSSREFS

Cf. A104272, A164368, A164288.
Sequence in context: A025584 A242256 A189483 * A157977 A105903 A045338
Adjacent sequences: A164949 A164950 A164951 * A164953 A164954 A164955


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Sep 01 2009


EXTENSIONS

Extended and edited by T. D. Noe, Nov 22 2010
Comments edited by Jonathan Sondow, Aug 27 2011


STATUS

approved



