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A198469
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a(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly seven primes.
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16
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1129, 113, 139, 139, 23, 47, 7, 7, 37, 67, 67, 37, 127, 3, 3, 5, 41, 11, 17, 5, 5, 5, 29, 71, 11, 101, 2, 2, 2, 101, 107, 2, 2, 71, 71, 191, 191, 227, 239, 281, 2, 197, 227, 107, 29, 569, 281, 821, 599, 1031, 521, 641, 659, 1229, 569, 1061, 1481, 2657, 641
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OFFSET
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2,1
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COMMENTS
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The sequence is unbounded.
Conjecture. In the supposition that there are infinitely many twin primes, every term beginning with the 15th is 2 or in A001359 (lesser of twin primes).
A generalization. Consider sequence A_r: "A_r(n) is the smallest prime(m) such that the interval (prime(m)*n, prime(m+1)*n) contains exactly r>=0 primes". Then the sequence is unbounded and, in the supposition that there are infinitely many twin primes, we conjecture that there exists number N=N(r)>=2, such that for n>=N every term A_r(n) is 2 or in A001359.
Proof of unboundedness. If the sequence is bounded,then for some k and for arbitrary n, a(n) is in the set {p_1,p_2,...,p_k}, where p_i=prime(i). This means that for all n there exists p_i<=p_k such that interval (n*p_i, n*p_(i+1)) contains exactly r primes. However, from the PNT it evidently follows that pi(n*p_(i+1))-pi(n*p_i) tends to infinity as n goes to infinity, i.e., for sufficiently large n we obtain a contradiction for every i<=k.
Note that from these arguments it follows more: for every fixed r>=0, A_r(n) tends to infinity as n goes to infinity. Thus a fixed prime which is in the sequence can repeat only a finite number of times.
In addition, note that the condition "in the supposition that there are infinitely many twin primes" means that, if after a large number N_(tw) there are no twin primes, then this sequence, existing after N_(tw), of course, cannot have any term in A001359.
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LINKS
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EXAMPLE
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Let n=20, and consider intervals of the form (20*prime(m), 20*prime(m+1)).
For 2, 3, 5, ..., the intervals (40,60), (60,100), (100,140), (140,220), (220,260), (260,340), (340,380), ... contain 5, 8, 9, 13, 8, 13, 7, ... primes. Hence the smallest such prime is 17.
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MATHEMATICA
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a[n_] := Catch[ For[m = 1, True, m++, p = Prime[m]; If[PrimePi[n*Prime[m + 1]] - PrimePi[n*p] == 7, Throw[p]]]]; Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Jan 18 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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