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A058188
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Number of primes between prime(n) and prime(n) + sqrt(prime(n)), where prime(n) is the n-th prime.
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3
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1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 0, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 1, 3, 4, 3, 2, 2, 1, 2, 3, 3, 4, 3, 3, 2, 1, 1, 3, 2, 1, 1, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 4, 3, 4, 3, 3, 4, 4, 3, 3, 2, 2, 3, 4, 3, 3, 3, 2, 2, 1, 3
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OFFSET
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1,12
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COMMENTS
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Conjecture: if prime(n)>=127, there is always at least one prime between prime(n) and prime(n) + sqrt(prime(n)). Easily checked for prime(n)<1.1e15 in existing maximal gap tables
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REFERENCES
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R. K. Guy: Unsolved problems in number theory, 2nd ed., Springer-Verlag,1994; Sections A8, A 9.
Paulo Ribenboim: The little book of big primes, Springer-Verlag,1991; 142ff
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LINKS
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EXAMPLE
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a(12) = 2 because between p(12)= 37 and 37+sqrt(37) = 43.08 there are two primes: 41 and 43
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MATHEMATICA
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Table[PrimePi[p+Sqrt[p]]-PrimePi[p], {p, Prime[Range[100]]}] (* Harvey P. Dale, Mar 13 2023 *)
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PROG
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(PARI) a(n) = my(p=prime(n)); primepi(p+sqrtint(p)) - n; \\ Michel Marcus, Jun 21 2017
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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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