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A053000
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a(n) = (smallest prime > n^2) - n^2.
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20
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2, 1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 5, 4, 1, 2, 1, 4, 7, 6, 1, 2, 3, 12, 1, 6, 1, 4, 3, 12, 7, 6, 7, 2, 7, 4, 1, 4, 3, 2, 1, 12, 13, 12, 13, 2, 13, 4, 5, 10, 3, 8, 3, 10, 1, 12, 1, 2, 7, 10, 7, 6, 3, 20, 3, 4, 1, 4, 13, 22, 3, 10, 5, 4, 1, 14, 3, 10, 5, 6, 21, 2, 9, 10, 1, 4, 15, 4, 9, 6, 1, 6, 3, 14
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OFFSET
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0,1
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COMMENTS
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Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.
Conjecture: a(n) <= 1+phi(n) = 1+A000010(n), for n>0. This improves on Oppermann's conjecture, which says a(n) < n. - Jianglin Luo, Sep 22 2023
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REFERENCES
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J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
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LINKS
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FORMULA
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MAPLE
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MATHEMATICA
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nxt[n_]:=Module[{n2=n^2}, NextPrime[n2]-n2]
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PROG
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(Python)
from sympy import nextprime
def a(n): nn = n*n; return nextprime(nn) - nn
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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