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A290564
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Number of primes between n^2 and 2*n^2.
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2
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1, 2, 3, 5, 6, 9, 10, 13, 15, 21, 23, 27, 29, 33, 39, 43, 45, 52, 56, 61, 67, 71, 78, 85, 90, 95, 102, 110, 117, 124, 131, 137, 145, 153, 163, 167, 180, 190, 196, 201, 211, 218, 233, 241, 252, 261, 271, 281, 290, 302, 314, 320, 329, 344, 355, 371, 385, 393, 407, 416, 423, 443
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OFFSET
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1,2
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COMMENTS
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a(n) >= 1 for n >= 1 (Bertrand's postulate).
(2*n^2)/log(2*n^2) + (0.992*2*n^2)/log^2(2*n^2) - n^2/log(n^2) - (1.2762*n^2)/log^2(n^2) < a(n) < (2*n^2)/log(2*n^2) + (1.2762*2*n^2)/log^2(2*n^2) - n^2/log(n^2) - (0.992*n^2)/log^2(n^2) (for n^2 > 599 - see at LINKS for Dusart) holds for a(n) except n = {1,2,4,10}.
It seems a(n) is strictly monotonically increasing.
It seems that lim_{n->inf} a(n)/A(n) == 1, with A(n) = (2*n^2)/log(2*n^2) + (2*n^2)/log^2(2*n^2)) - n^2/log(n^2) - n^2/log(n^2).
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LINKS
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FORMULA
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a(n) = pi(2n^2) - pi(n^2).
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MAPLE
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MATHEMATICA
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Table[PrimePi[2 n^2] - PrimePi[n^2], {n, 1, 100}]
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PROG
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(PARI) a(n) = primepi(2*n^2) - primepi(n^2); \\ Michel Marcus, Aug 06 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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