

A002331


Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n).
(Formerly M0096 N0033)


23



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

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. 10111013.
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), 517528.
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^2a^2 = A070079,
with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2a^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, n1] (* JeanFrançois Alcover, Feb 26 2016 *)


PROG

(PARI) f(p)=my(s=lift(sqrt(Mod(1, p))), x=p, t); if(s>p/2, s=ps); while(s^2>p, t=s; s=x%s; x=t); s
forprime(p=2, 1e3, if(p%43, print1(sqrtint(pf(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%43, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013


CROSSREFS

Cf. A002330, A002313, A002144.
Sequence in context: A051493 A338201 A029173 * A060805 A184342 A030767
Adjacent sequences: A002328 A002329 A002330 * A002332 A002333 A002334


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



