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A179076
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Number of primes of the form k^2 + 1 less than n.
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1
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0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5
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OFFSET
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1,6
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COMMENTS
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The first n such that a(n) = 5 is a(102). Records in a(n) are a(n) = A002496(n)+1. Hardy and Littlewood conjectured that, asymptotically, a(n) ~ c*(sqrt(n))/log n, where c ~ 1.3727.
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 2nd Edn., Springer, 1994, A1, pp.4-5.
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LINKS
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EXAMPLE
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a(3) = 1 because the unique prime of the form k^2 + 1 less than 3 is 1^2 + 1 = 2. The smallest value of n to reach the next record is a(6) = 2 because a(18) = 2, the two primes of the form k^2 + 1 less than 6 are 2 and 2^2 + 1 = 5. The smallest value of n to reach the next record is a(18) = 3 because the three primes of the form k^2 + 1 less than 18 are 2, 5, and 4^2 + 1 = 17.
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MATHEMATICA
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Accumulate[Table[If[PrimeQ[n]&&IntegerQ[Sqrt[n-1]], 1, 0], {n, 0, 120}]] (* Harvey P. Dale, Jun 22 2024 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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