A002331 Values of x in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n). (Formerly M0096 N0033)

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

Offset

1,3

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A. T. Benjamin and D. Zeilberger, Pythagorean primes and palindromic continued fractionsINTEGERS 5(1) (2005) #A30

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

a(n+1) = Min(A002972(n), 2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010

a(n) = A363051(A002313(n)). - R. J. Mathar, Jan 31 2024

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.
# alternative
A002331 := proc(n)
    A363051(A002313(n)) ;
end proc:
seq(A002331(n), n=1..100) ; # R. J. Mathar, Feb 01 2024

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-François 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, A027862 (locates y=x+1).

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