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A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0096 N0033)
22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n+1) = MIN(A002972(n), 2*A002973(n)). [Reinhard Zumkeller, Feb 16 2010]

REFERENCES

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).

LINKS

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.

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

K. Matthews, Serret's algorithm Server

J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem

FORMULA

a(n) = A096029(n) - A096030(n) for n>1. - Lekraj Beedassy, Jul 16 2004

EXAMPLE

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

.................................

MAPLE

See A002330 for Maple program.

MATHEMATICA

pmax = 1000; x[p_] := Module[{x, y}, x /. ToRules[Reduce[0 <= x <= y && x^2 + y^2 == p, {x, y}, Integers]]]; For[n=1; p=2, p<pmax, p = NextPrime[p], If[Mod[p, 4] == 1 || Mod[p, 4] == 2, a[n] = x[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* Jean-Fran├žois Alcover, Feb 26 2016 *)

PROG

(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

(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[2]

forprime(p=2, 1e3, if(p%4-3, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013

CROSSREFS

Cf. A002330, A002313, A002144.

Sequence in context: A029196 A051493 A029173 * A060805 A184342 A030767

Adjacent sequences:  A002328 A002329 A002330 * A002332 A002333 A002334

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified May 24 21:44 EDT 2016. Contains 273253 sequences.