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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 k-th 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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]

All Ramanujan 3-primes 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

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Last modified October 23 07:11 EDT 2019. Contains 328336 sequences. (Running on oeis4.)