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A127308
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Number of ways of writing the n-th prime prime(n) as a sum of 24 squares.
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1
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1104, 16192, 1362336, 44981376, 6631997376, 41469483552, 793229226336, 2697825744960, 22063059606912, 282507110257440, 588326886375936, 4119646755044256, 12742799887509216, 21517654506205632, 57242599902057216
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OFFSET
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1,1
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COMMENTS
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|a(n) - (16/691)*(prime(n)^11 + 1)| <= (66304/691)*sqrt(prime(n)^11) (proved by Deligne).
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 107.
Barry Mazur, Controlling our errors, Nature 443, 7 (2006) 38-40.
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LINKS
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FORMULA
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a(n) ~ (16/691)*(prime(n)^11 + 1) as n -> oo.
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EXAMPLE
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For prime(1) = 2, two of the 24 squares are (+-1)^2 and the other 22 are 0^2, so a(1) = 2*2*binomial(24,2) = 4*276 = 1104.
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MATHEMATICA
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Table[SquaresR[24, Prime[n]], {n, 1, 70}]
Table[Abs[16/691 (p^11 + 1) + 33152/691 RamanujanTau[p]], {p, Prime@Range@70}] (* Giorgos Kalogeropoulos, Dec 15 2022 *)
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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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