A002330 Value of y in the solution to p = x^2 + y^2, x <= y, with prime p = A002313(n). (Formerly M0462 N0169)

1, 2, 3, 4, 5, 6, 5, 7, 6, 8, 8, 9, 10, 10, 8, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20, 25, 22, 26, 28, 25

Offset

1,2

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)

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) + 1, for n>1. - Lekraj Beedassy, Jul 21 2004

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

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

a := []; for x from 0 to 50 do for y from x to 50 do p := x^2+y^2; if isprime(p) then a := [op(a), [p, x, y]]; fi; od: od: writeto(trans); for i from 1 to 158 do lprint(a[i]); od: # then sort the triples in "trans"

Mathematica

Flatten[#, 1]&[Table[PowersRepresentations[Prime[k], 2, 2], {k, 1, 142}]][[All, 2]] (* Jean-François Alcover, Jul 05 2011 *)

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(f(p)", "))) \\ Charles R Greathouse IV, Apr 24 2012
(PARI) do(p)=qfbsolve(Qfb(1, 0, 1), p)[1]
forprime(p=2, 1e3, if(p%4-3, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013
(PARI) print1(1); forprimestep(p=5, 1e3, 4, print1(", "qfbcornacchia(1, p)[1])) \\ Charles R Greathouse IV, Sep 15 2021

Crossrefs

Cf. A002331, A002313, A002144.

Sequence in context: A245352 A099033 A187786 * A335130 A358747 A358230

Adjacent sequences: A002327 A002328 A002329 * A002331 A002332 A002333

Keyword

nonn

Author

N. J. A. Sloane

Status

approved