A002330 -id:A002330 - cp's OEIS frontend

A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

3, 5, 15, 21, 35, 9, 45, 11, 55, 39, 65, 99, 91, 15, 105, 51, 85, 165, 19, 95, 195, 221, 105, 209, 255, 69, 115, 231, 285, 25, 75, 175, 299, 225, 275, 189, 325, 399, 391, 29, 145, 351, 425, 261, 459, 279, 341, 165, 231, 575, 465, 551, 35, 105, 609, 315, 589, 385, 675

Offset: 1

Lekraj Beedassy, May 06 2002

Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.

Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - Lekraj Beedassy, Jul 22 2005

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

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002144, A002330, A002331, A080109, A070151

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

a(n)=A079886(n)*A079887(n) - Benoit Cloitre, Jan 13 2003

a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the Lekraj Beedassy, comment. - Wolfdieter Lang, Jan 13 2015

More terms from Benoit Cloitre, Jan 13 2003

Edited: Used a different name and moved old name to the comment section. - Wolfdieter Lang, Jan 13 2015

A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

Original entry on oeis.org

0, 2, 1, 4, 2, 5, 4, 7, 8, 5, 2, 7, 10, 1, 10, 8, 2, 7, 4, 13, 1, 14, 8, 14, 11, 7, 14, 13, 16, 8, 11, 16, 17, 7, 2, 19, 4, 17, 19, 11, 1, 14, 5, 10, 22, 16, 4, 23, 20, 8, 23, 13, 10, 5, 16, 22, 20, 19, 25, 4, 11, 22, 25, 8, 26, 13, 1, 28, 28, 26, 23, 29, 28

Offset: 1

N. J. A. Sloane

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).
B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Cf. A001480, A007645, A000196, A010052, A033428.

a001479 n = a000196 $ head $
   filter ((== 1) . a010052) $ map (a007645 n -) $ tail a033428_list
-- Reinhard Zumkeller, Jul 11 2013

nmax = 56; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := x /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 0; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[1])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Definition revised by N. J. A. Sloane, Jan 29 2013

A001480 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = y.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 1, 4, 5, 4, 1, 6, 3, 5, 7, 6, 7, 2, 8, 1, 7, 3, 6, 8, 5, 6, 3, 9, 8, 5, 4, 10, 11, 2, 11, 6, 4, 10, 12, 9, 12, 11, 1, 9, 13, 2, 7, 13, 4, 12, 13, 14, 11, 7, 9, 10, 4, 15, 14, 9, 6, 15, 5, 14, 16, 1, 3, 7, 10, 2, 5, 14, 17, 13, 9, 16, 17

Offset: 1

N. J. A. Sloane

a(n) = A000196((A007645(n) - A000290(A001479(n))) / 3). - Reinhard Zumkeller, Jul 11 2013

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).
B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Cf. A001479, A007645.

a001480 n = a000196 $ (`div` 3) $ (a007645 n) - (a001479 n) ^ 2
-- Reinhard Zumkeller, Jul 11 2013

nmax = 63; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := y /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 1; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[2])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Definition revised by N. J. A. Sloane, Jan 29 2013

A122921 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, minimizing the value of x. a(n) is that x.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 5, 4, 4, 4, 4, 5, 4, 5, 5, 4, 4, 4, 5, 4, 6, 5, 5, 6, 4, 5, 5, 5, 5, 6, 5, 4, 6, 5, 5, 5, 6, 5, 6, 6, 5, 6, 5, 5, 6, 6, 5, 6, 6, 5, 7, 5, 6, 6, 6, 6, 6, 6, 5, 6, 6, 7, 6, 8, 6, 6, 7, 5, 6, 6, 7, 6

Offset: 0

Franklin T. Adams-Watters, Sep 19 2006

nonn

			10 = 2^2 + 2^2 + 1^2 + 1^2, so a(10) = 2. The only representation for 11 is 3^2 + 1^2 + 1^2 + 0^2, so a(11) = 3.

David Consiglio, Jr., Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python program

Cf. A122922, A122923, A122924, A122925, A122926, A122927, A002330.

Analogs for 3 squares: A261904 and A261915.

A002334 Least positive integer x such that prime A038873(n) = x^2 - 2y^2 for some y.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 7, 11, 9, 9, 11, 13, 11, 11, 15, 13, 13, 13, 17, 15, 19, 15, 19, 17, 21, 17, 19, 17, 17, 19, 21, 25, 19, 19, 23, 25, 23, 21, 23, 21, 21, 29, 23, 25, 23, 27, 29, 23, 31, 33, 25, 29, 27, 25, 25, 27, 29, 35, 31, 31, 27, 29, 33, 31, 29, 29, 29, 29, 37, 31, 41, 35

Offset: 1

N. J. A. Sloane

A prime p is representable in the form x^2 - 2y^2 iff p is 2 or p == 1 or 7 (mod 8). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
From Wolfdieter Lang, Feb 17 2015: (Start)
For the corresponding y terms see A002335.
a(n), together with A002335(n), gives the fundamental positive  solution of the first class of this (generalized) Pell equation. The prime 2 has only one class of proper solutions. The fundamental positive solutions of the second class for the primes from A001132 are given in A254930 and A254931. (End)

			The first solutions [x(n), y(n)] are (the prime is given as first entry): [2,[2,1]], [7,[3,1]], [17,[5,2]], [23,[5,1]], [31,[7,3]], [41,[7,2]], [47,[7,1]], [71,[11,5]], [73,[9,2]], [79,[9,1]], [89,[11,4]], [97,[13,6]], [103,[11,3]], [113,[11,2]], [127,[15,7]], [137,[13,4]], [151,[13,3]], [167,[13,1]], [191,[17,7]], [193,[15,4]], [199,[19,9]], [223,[15,1]], [233,[19,8]], [239,[17,5]], [241,[21,10]], [257,[17,4]], [263,[19,7]], [271,[17,3]], ... - _Wolfdieter Lang_, Feb 17 2015

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

Robert Israel, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Cf. A002335, A035251.

with(numtheory): readlib(issqr): for i from 1 to 250 do p:=ithprime(i): pmod8:=modp(p,8): if p=2 or pmod8=1 or pmod8=7 then for y from 1 do x2:=p+2*y^2: if issqr(x2) then printf("%d,",sqrt(x2)): break fi od fi od: # Pab Ter, May 08 2004

maxPrimePi = 200;
Reap[Do[If[MatchQ[Mod[p, 8], 1|2|7], rp = Reduce[x > 0 && y > 0 && p == x^2 - 2*y^2, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x0 = xy[[All, 1]] // Min // Simplify; Print[{p, xy[[1]]} ]; Sow[x0]]], {p, Prime[Range[maxPrimePi]]}]][[2, 1]] (* Jean-François Alcover, Oct 27 2019 *)

a(n)^2 - 2*A002335(n)^2 = A038873(n), n >= 1, and a(n) is the least positive integer satisfying this Pell type equation. - Wolfdieter Lang, Feb 12 2015

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

The name has been changed in order to be more precise and to conform with A002335. The offset has been changed to 1. - Wolfdieter Lang, Feb 12 2015

A002335 Least positive integer y such that A038873(n) = x^2 - 2y^2 for some x.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 2, 1, 4, 6, 3, 2, 7, 4, 3, 1, 7, 4, 9, 1, 8, 5, 10, 4, 7, 3, 2, 5, 8, 12, 2, 1, 9, 11, 8, 4, 7, 2, 1, 14, 6, 9, 5, 11, 13, 2, 14, 16, 4, 11, 8, 3, 2, 7, 10, 17, 12, 11, 1, 7, 13, 10, 6, 4, 3, 1, 16, 7, 20, 13, 5, 15, 4, 12, 2, 21, 14, 11, 7, 16, 13, 18, 5, 20, 9, 1, 8, 17, 14

Offset: 1

N. J. A. Sloane

A prime p is representable in the form x^2-2y^2 iff p is 2 or p == 1 or 7 mod 8. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

Robert Israel, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Cf. A002334, A035251.

with(numtheory): readlib(issqr):for i from 1 to 300 do p:=ithprime(i): pmod8:=modp(p,8): if p=2 or pmod8=1 or pmod8=7 then for y from 1 do if issqr(p+2*y^2) then printf("%d,",y): break fi od fi od: # Pab Ter, Oct 22 2005

maxPrimePi = 200;
Reap[Do[If[MatchQ[Mod[p, 8], 1|2|7], rp = Reduce[x > 0 && y > 0 && p == x^2 - 2*y^2, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y0 = xy[[All, 2]] // Min // Simplify; Print[{p, xy[[1]]} ]; Sow[y0]]], {p, Prime[Range[maxPrimePi]]}]][[2, 1]] (* Jean-François Alcover, Oct 27 2019 *)

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

A002332 Numbers x such that p = x^2 + 2y^2, with prime p = A033203(n).

Original entry on oeis.org

0, 1, 3, 3, 1, 3, 5, 3, 7, 1, 9, 9, 5, 3, 9, 9, 3, 11, 1, 9, 11, 7, 15, 15, 13, 3, 15, 9, 11, 17, 5, 13, 7, 3, 15, 19, 3, 11, 9, 19, 21, 21, 13, 15, 21, 7, 3, 19, 23, 15, 21, 11, 17, 3, 9, 23, 15, 13, 21, 25, 9, 5, 21, 23, 17, 27, 11, 25, 3, 19, 27, 27, 29, 9, 1, 5, 27, 17, 15, 21, 27

Offset: 1

N. J. A. Sloane

nonn

For p>2, x and y are uniquely determined [Frei, Th. 3]. - N. J. A. Sloane, May 30 2014

The corresponding y numbers are given in A002333.

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55-58 and 64.
D. S., Review of A Table of Primes of Z[(-2)^(1/2)] by J. H. Jordan and J. R. Rabung, Math. Comp., 23 (1969), p. 458.

Cf. A002333.

f[ p_ ] := For[ y=1, True, y++, If[ IntegerQ[ x=Sqrt[ p-2y y ] ], Return[ x ] ] ]; f/@Select[ Prime/@Range[ 1, 200 ], Mod[ #, 8 ]<4& ]

More terms from Dean Hickerson, Oct 07 2001

A079887 Values of y-x where p runs through the primes of form 4k+1 and p=x^2+y^2, 0<=x<=y.

Original entry on oeis.org

1, 1, 3, 3, 5, 1, 5, 1, 5, 3, 5, 9, 7, 1, 7, 3, 5, 11, 1, 5, 13, 13, 5, 11, 15, 3, 5, 11, 15, 1, 3, 7, 13, 9, 11, 7, 13, 19, 17, 1, 5, 13, 17, 9, 17, 9, 11, 5, 7, 23, 15, 19, 1, 3, 21, 9, 19, 11, 25, 21, 7, 25, 17, 1, 13, 5, 15, 23, 11, 17, 5, 25, 23, 9, 3, 5, 19, 15, 27, 25, 13, 1, 19, 29, 27

Offset: 1

Benoit Cloitre, Jan 13 2003

nonn

Also values of x where p runs through the primes of form 4k+1 and 2*p=x^2+y^2, 0<=xColin Barker, Jul 07 2014

Cf. A002330, A002331, A002313, A002144, A079886.

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y - x /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A079887 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

a(n) = A002330(n+1)-A002331(n+1). - R. J. Mathar, Jan 09 2017

A002333 Numbers y such that p = x^2 + 2y^2, with prime p = A033203(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 3, 5, 3, 6, 1, 2, 6, 7, 4, 5, 8, 3, 9, 7, 6, 9, 1, 2, 6, 11, 4, 10, 9, 3, 12, 9, 12, 13, 8, 3, 14, 12, 13, 6, 1, 2, 12, 11, 5, 15, 16, 9, 3, 13, 8, 15, 12, 17, 16, 6, 14, 15, 10, 3, 17, 18, 11, 9, 15, 4, 18, 9, 20, 15, 7, 8, 3, 20, 21, 21, 10, 18, 19, 16, 11, 22, 18

Offset: 1

N. J. A. Sloane

nonn

The corresponding x numbers are given in A002332.

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904. [Annotated scans of selected pages]
D. S., Review of A Table of Primes of Z[(-2)^(1/2)] by J. H. Jordan and J. R. Rabung, Math. Comp., 23 (1969), p. 458.

Cf. A002332.

g[p_] := For[y=1, True, y++, If[IntegerQ[Sqrt[p-2y y]], Return[y]]]; g/@Select[Prime/@Range[1, 200], Mod[ #, 8]<4&]

More terms from Dean Hickerson, Oct 07 2001

A079886 Values of x+y where p runs through the primes of form 4k+1 and p=x^2+y^2, 0<=x<=y.

Original entry on oeis.org

3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 11, 13, 15, 15, 17, 17, 15, 19, 19, 15, 17, 21, 19, 17, 23, 23, 21, 19, 25, 25, 25, 23, 25, 25, 27, 25, 21, 23, 29, 29, 27, 25, 29, 27, 31, 31, 33, 33, 25, 31, 29, 35, 35, 29, 35, 31, 35, 27, 31, 37, 29, 35, 39, 37, 39, 37, 33, 39, 37, 41, 33

Offset: 1

Benoit Cloitre, Jan 13 2003

nonn

Also values of y where p runs through the primes of form 4k+1 and 2*p=x^2+y^2, 0 Colin Barker, Jul 07 2014

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002330, A002331, A002313, A002144, A079887.

N:= 100: # to get values corresponding to primes <= 4*N+1
P:= select(isprime, [seq(4*i+1,i=0..N)]):
F:= proc(p) local f; f:= GaussInt:-GIfactors(p)[2][1][1]; abs(Re(f))+abs(Im(f)) end proc:
map(F,P); # Robert Israel, Jul 07 2014

pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := x + y /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A079886 = f /@ pp (* Jean-François Alcover, Jan 15 2015 *)

cp's OEIS Frontend

A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

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A001480 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = y.

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A122921 Express n as the sum of four squares, x^2+y^2+z^2+w^2, x>=y>=z>=w>=0, minimizing the value of x. a(n) is that x.

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A002334 Least positive integer x such that prime A038873(n) = x^2 - 2y^2 for some y.

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Maple

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A002335 Least positive integer y such that A038873(n) = x^2 - 2y^2 for some x.

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Comments

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Crossrefs

Programs

Maple

Mathematica

Extensions

A002332 Numbers x such that p = x^2 + 2y^2, with prime p = A033203(n).

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