

A002330


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


39



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
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OFFSET

1,2


COMMENTS

Equals A096029(n) + A096030(n) + 1, for entries after the first.  Lekraj Beedassy, Jul 21 2004
a(n+1) = MAX(A002972(n), 2*A002973(n)). [From 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


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

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]] (* JeanFrançois Alcover, Jul 05 2011 *)


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(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%43, print1(do(p)", "))) \\ Charles R Greathouse IV, Sep 26 2013


CROSSREFS

Cf. A002331, A002313, A002144.
Sequence in context: A245352 A099033 A187786 * A305900 A287943 A305211
Adjacent sequences: A002327 A002328 A002329 * A002331 A002332 A002333


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



