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A002331
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Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0096 N0033)
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21
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1, 1, 2, 1, 2, 1, 4, 2, 5, 3, 5, 4, 1, 3, 7, 4, 7, 6, 2, 9, 7, 1, 2, 8, 4, 1, 10, 9, 5, 2, 12, 11, 9, 5, 8, 7, 10, 6, 1, 3, 14, 12, 7, 4, 10, 5, 11, 10, 14, 13, 1, 8, 5, 17, 16, 4, 13, 6, 12, 1, 5, 15, 2, 9, 19, 12, 17, 11, 5, 14, 10, 18, 4, 6, 16, 20, 19, 10, 13, 4, 6, 15, 22, 11, 3, 5
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OFFSET
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1,3
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COMMENTS
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a(n+1) = MIN(A002972(n), 2*A002973(n)). [From Reinhard Zumkeller, Feb 16 2010]
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REFERENCES
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A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.
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LINKS
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T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Noe)
John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), pp. 1011-1013.
K. Matthews, Serret's algorithm Server
Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem
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FORMULA
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Equals A096029(n)-A096030(n) for entries after the first. - Lekraj Beedassy, Jul 16 2004
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EXAMPLE
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The following table shows the relationship
between several closely related sequences:
Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
a = A002331, b = A002330, t_1 = ab/2 = A070151;
p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
---------------------------------
.p..a..b..t_1..c...d.t_2.t_3..t_4
---------------------------------
.5..1..2...1...3...4...4...3....6
13..2..3...3...5..12..12...5...30
17..1..4...2...8..15...8..15...60
29..2..5...5..20..21..20..21..210
37..1..6...3..12..35..12..35..210
41..4..5..10...9..40..40...9..180
53..2..7...7..28..45..28..45..630
.................................
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MAPLE
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See A002330 for Maple program.
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PROG
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(PARI) f(p)=my(s=lift(sqrt(Mod(-1, p))), x=p, t); if(s>p/2, s=p-s); while(s^2>p, t=s; s=x%s; x=t); s
forprime(p=2, 1e3, if(p%4-3, print1(sqrtint(p-f(p)^2)", ")))
\\ Charles R Greathouse IV, Apr 24 2012
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CROSSREFS
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Cf. A002330, A002313, A002144.
Sequence in context: A029196 A051493 A029173 * A060805 A184342 A030767
Adjacent sequences: A002328 A002329 A002330 * A002332 A002333 A002334
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KEYWORD
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nonn,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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