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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 2
2, 3, 5, 13, 19, 29, 31, 43, 47, 61, 67, 73, 79, 83, 101, 103, 107, 109, 137, 139, 151, 157, 167, 173, 181, 193, 197, 199, 229, 233, 241, 257, 263, 271, 277, 281, 283, 313, 317, 349, 353, 359, 367, 373, 379, 389, 401, 409, 431, 433, 439, 443, 461, 463, 467, 487, 499 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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]

All 3-Labos primes are in this sequence.

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

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.

MATHEMATICA

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]

CROSSREFS

Cf. A104272, A080359, A164952, A164368, A164288, A164294

Sequence in context: A157976 A155738 A215371 * A145908 A173830 A224223

Adjacent sequences:  A164955 A164956 A164957 * A164959 A164960 A164961

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Sep 02 2009

EXTENSIONS

Extended by T. D. Noe, Nov 23 2010

STATUS

approved

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Last modified July 28 10:34 EDT 2014. Contains 244989 sequences.