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A218831
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a(n) is the least r > 1 for which the interval (r*n, r*(n+1)) contains no prime, or a(n)=0 if no such r exists.
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6
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0, 0, 0, 2, 0, 4, 2, 3, 0, 2, 3, 2, 2, 0, 6, 2, 2, 3, 2, 6, 3, 2, 4, 2, 2, 7, 2, 2, 4, 3, 2, 2, 4, 2, 4, 4, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 2, 3, 4, 2, 3, 2, 2, 2, 2, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 3, 2
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OFFSET
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1,4
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COMMENTS
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In the first 50000000 terms a(n) is 0 only for n=1, 2, 3, 5, 9, 14. In the same range the largest value of a(n) is 16 at n=2540, 77384, 1679690, 3240054, 13078899.
a(1)=0 is "Bertrand's postulate," which states that there is always a prime between k and 2*k. This was first proved by P. Chebyshev.
Note that the equations a(2) = a(3) = 0 are results of M. El. Buchraoui and A. Loo respectively and could be proved with the uniform positions, using Theorem 30 for generalized Ramanujan numbers from the Shevelev link. The equation a(5) = 0 follows from the result of J. Nagura. For proof of the equations a(9)=a(14)=0, we used a known result of L. Schoenfeld (1976) that states that for n>2010760, between n and n*(1+1/16597) there is always a prime.
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LINKS
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N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13
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FORMULA
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MATHEMATICA
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rmax = 100; a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[PrimePi[r*n] == PrimePi[r*(n + 1)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 87}] (* Jean-François Alcover, Dec 13 2012 *)
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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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