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Search: id:A127308
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| A127308 |
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Number of ways of writing the n-th prime p(n) as a sum of 24 squares. |
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+0
1
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| 1104, 16192, 1362336, 44981376, 6631997376, 41469483552, 793229226336, 2697825744960, 22063059606912, 282507110257440, 588326886375936, 4119646755044256, 12742799887509216, 21517654506205632, 57242599902057216
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Deligne proved that |a(n) - (16/691)*(p(n)^11 + 1)| <= (66304/691)*sqrt(p(n)^11).
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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, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture, Current Events Bulletin, Amer. Math. Soc., 2007.
Barry Mazur, Controlling our errors, Nature 443, 7 (2006) 38-40.
Barry Mazur, Finding meaning in error terms, Bull. Amer. Math. Soc., 45 (No. 2, 2008), 185-228.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..70
Barry Mazur, The Structure of Error Terms in Number Theory and an Introduction to the Sato-Tate Conjecture
Barry Mazur, Controlling our errors
Tony Phillips, Math in the Media
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FORMULA
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a(n) ~ (16/691)*(p(n)^11 + 1) as n -> oo.
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EXAMPLE
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For p(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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Needs["NumberTheory`NumberTheoryFunctions`"]; Table[SumOfSquaresR[24, Prime[n]], {n, 1, 70}]
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CROSSREFS
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a(n) = A000156(p(n)).
Cf. A000594.
Adjacent sequences: A127305 A127306 A127307 this_sequence A127309 A127310 A127311
Sequence in context: A080317 A068279 A060519 this_sequence A022055 A008688 A107519
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KEYWORD
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nonn
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AUTHOR
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Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 10 2007
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