A031363 -id:A031363 - cp's OEIS frontend

A238240 Positive integers n such that x^2 - 20xy + y^2 + n = 0 has integer solutions.

18, 35, 50, 63, 72, 74, 83, 90, 95, 98, 99, 107, 140, 162, 171, 200, 215, 227, 252, 266, 275, 288, 296, 315, 332, 347, 359, 360, 362, 371, 380, 387, 392, 395, 396, 407, 428, 450, 491, 495, 530, 539, 560, 567, 602, 623, 626, 635, 648, 666, 684, 695, 711, 722, 743, 747, 755, 770, 791, 794, 800, 810

Offset: 1

Colin Barker, Feb 20 2014

nonn

Positive integers n such that x^2 - 99 y^2 + n = 0 has integer solutions. - Robert Israel, Oct 22 2024

			63 is in the sequence because x^2 - 20xy + y^2 + 63 = 0 has integer solutions, for example (x, y) = (1, 16).

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

Cf. A075839 (n = 18), A221763 (n = 63), A198947 (n = 90), A001085 (n = 99).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331.

filter:= t -> [isolve(99*y^2 - z^2 = t)] <> []:
select(filter, [$1..1000]); # Robert Israel, Oct 22 2024

Corrected by Robert Israel, Oct 22 2024

A031364 Number of coincidence site modules of index 10n+1 in an icosahedral module.

Original entry on oeis.org

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 20, 0, 0, 40, 30, 0, 0, 0, 0, 30, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 80, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0

Offset: 1

N. J. A. Sloane

a(n) is nonzero iff n is of the form x^2+x*y+y^2 (A031363).

Michael Baake, "Solution of coincidence problem in dimensions d<=4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 (2006), Prop. 5.4.
Michael Baake and Peter AB Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 716.
Sean A. Irvine, Java program (github)

Cf. A031363.

Dirichlet series: ((1+5^(-s))/(1-5^(1-s))) * Product_{p = +-2 (mod 5)} ((1+p^(-2*s))/(1-p^(2*(1-s)))) * Product_{p = +-1 (mod 5)} ((1+p^(-s))/(1-p^(1-s)))^2. - Sean A. Irvine, Apr 29 2020

Missing a(8)=0 and more terms from Sean A. Irvine, Apr 29 2020

A263850 Let R = Z[(1+sqrt(5))/2] denote the ring of integers in the real quadratic number field of discriminant 5. Then a(n) is the largest integer k such that every totally positive element of norm n in R can be written as a sum of three squares in R in at least k ways, or 0 if there is no totally positive element of norm n.

Original entry on oeis.org

1, 6, 0, 0, 12, 24, 0, 0, 0, 32, 0, 24, 0, 0, 0, 0, 54, 0, 0, 24, 24, 0, 0, 0, 0, 30, 0, 0, 0, 24, 0, 48, 0, 0, 0, 0, 48, 0, 0, 0, 0, 96, 0, 0, 24, 48, 0, 0, 0, 96, 0, 0, 0, 0, 0, 48, 0, 0, 0, 24, 0, 120, 0, 0, 108, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 72, 0, 0, 48, 120, 54, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 18 2015

nonn

The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015. See A263849 for further information.

Note that there are examples of totally positive elements x and y in R which have the same norm, but for which the number of ways of writing x as a sum of three squares in R is different to the number of ways of writing y as a sum of three squares in R. See A263849 for explicit examples. - Robin Visser, Mar 28 2025

Maass, Hans. Über die Darstellung total positiver Zahlen des Körpers R (sqrt(5)) als Summe von drei Quadraten, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1, pp. 185-191, 1941.

Robin Visser, Table of n, a(n) for n = 0..10000
Harvey Cohn, A computation of some bi-quadratic class numbers, Math. Tables Aids Comput. 12 (1958), 213-217. See pages 215-217.

Cf. A263849 (another version of this sequence), A031363, A035187.

Cf. A005875 (sum of 3 squares in Z), A000118 (sum of 4 squares in Z).

import itertools
def a(n):
    if n==0: return 1
    if any([((r[0]%5 in [2,3]) and (r[1]%2==1)) for r in factor(n)]): return 0
    K. = NumberField(x^2-x-1); cw = w.coordinates_in_terms_of_powers(); ans = []
    for idl in K.ideals_of_bdd_norm(n)[n]:
        for u in [1,-1,w,-w]:
            X,Y = cw(u*idl.gens_reduced()[0]); num = 0
            if (X < 0): continue
            for b in range(-isqrt(X), isqrt(X)+1):
                for d in range(-isqrt(X-b^2), isqrt(X-b^2)+1):
                    for f in range(-isqrt(X-b^2-d^2), isqrt(X-b^2-d^2)+1):
                        S = b^2+d^2+f^2; M = isqrt(X-S);
                        for (a,c,e) in itertools.product(range(-M, M+1), repeat=3):
                            if (a^2+c^2+e^2+S==X) and (2*(a*b+c*d+e*f)+S==Y): num += 1
            if (num > 0): ans.append(num)
    return min(ans)  # Robin Visser, Mar 28 2025

A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.

Original entry on oeis.org

0, 1, 4, 9, 11, 16, 23, 25, 36, 37, 44, 49, 53, 64, 67, 71, 81, 91, 92, 99, 100, 113, 119, 121, 133, 137, 144, 148, 163, 169, 176, 179, 191, 196, 207, 212, 221, 225, 247, 253, 256, 268, 275, 284, 287, 289, 317, 323, 324, 331, 333, 361, 364, 368, 379, 389, 396, 400, 401, 407, 421, 427, 441, 443, 449

Offset: 1

Gheorghe Coserea, Jul 29 2018

nonn

Discriminant 77.

In general, for k>=0 the positive part of the set S = {x^2 - k*x*y + y^2: x,y in Z} is given by the numbers of the form x^2 + k*x*y + y^2 with 0 <= x <= y natural numbers.

Gheorghe Coserea, Table of n, a(n) for n = 1..100001

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), A031363(k=3), A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), this sequence.

seq(M,k=9) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
concat(0, seq(449))

A336403 Multiplicative closure of A045468: numbers which are the product of zero or more primes which are 1 or 4 mod 5.

Original entry on oeis.org

1, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, 121, 131, 139, 149, 151, 179, 181, 191, 199, 209, 211, 229, 239, 241, 251, 269, 271, 281, 311, 319, 331, 341, 349, 359, 361, 379, 389, 401, 409, 419, 421, 431, 439, 449, 451, 461, 479, 491, 499, 509

Offset: 1

David Friend, Jul 20 2020

The subsequence of A089270 which excludes terms divisible by 5.

David Friend, Table of n, a(n) for n = 1..10000
David Friend, Fibonacci-Like Sequences and their Cassini Constants, Multiplication and Factorisation

Cf. A031363 and its subset A089270 and its subset A038872 and its subset A045468.

Numbers of the form r^2 + 3*r*s + s^2 not divisible by 5, where r and s are relatively prime and r > s >= 0.

New name and general cleanup by Charles R Greathouse IV, Sep 09 2022

A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -43.

Original entry on oeis.org

1, 4, 9, 11, 13, 16, 17, 23, 25, 31, 36, 41, 43, 44, 47, 49, 52, 53, 59, 64, 67, 68, 79, 81, 83, 92, 97, 99, 100, 101, 103, 107, 109, 117, 121, 124, 127, 139, 143, 144, 153, 164, 167, 169, 172, 173, 176, 181, 187, 188, 193, 196, 197, 207, 208, 212, 221, 225

Offset: 1

N. J. A. Sloane

nonn

Also, positive numbers of the form x^2 + xy + 11y^2 (discriminant -43).

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A106891 (Primes of the form x^2 + xy + 11y^2).

Cf. A035147, A031363, A035256.

m=-43; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

More terms from Colin Barker, Jun 19 2014

A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

20, 39, 56, 71, 80, 84, 95, 104, 111, 116, 119, 120, 156, 180, 191, 224, 239, 255, 284, 296, 311, 320, 336, 351, 359, 380, 399, 404, 416, 431, 444, 455, 464, 471, 476, 479, 480, 500, 504, 551, 596, 599, 624, 639, 680, 695, 696, 719, 720, 756, 764, 791, 824

Offset: 1

Colin Barker, Feb 20 2014

nonn

			39 is in the sequence because x^2 - 22xy + y^2 + 39 = 0 has integer solutions, for example (x, y) = (2, 43).

Cf. A157014 (n = 20), A137881 (n = 104), A077422 (n = 120), A133275 (n = 180).

Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330, A236331, A238240.

A240469 Values k where the maximum number of distinct rational solutions to x^2 - Dy^2 = t, 0 < D <= k, 0 < t <= k, achieves a new record.

Original entry on oeis.org

1, 2, 7, 10, 17, 32, 73, 144, 241, 336, 360, 720, 1080, 1260

Offset: 1

Ralf Stephan, Apr 06 2014

Record values are in A240470.

			All Diophantine equations x^2 - Dy^2 = t, 0 < D <= 16, 0 < t <= 16, D squarefree, have fewer than 4 distinct solutions; the first with 4 solutions is x^2 - 17y^2 = 16 with the solutions (x,y) = (9/2,1/2), (21,5), (4,0), (13,3), so 17 is in the sequence.

Cf. A218459, A077428, A035251, A031363.

{ r(l,k)=if(!issquarefree(l)||!polisirreducible(z^2-l),return(0));v=bnfisintnorm(bnfinit(z^2-l), k);if(!#v,return(0));s=0;for(k=1,#v,p=v[k];a=polcoeff(p,0);b=polcoeff(p,1);f=1;for(l=k+1,#v,p=v[l];aa=polcoeff(p,0);bb=polcoeff(p,1);if(abs(a)==abs(aa)&&abs(b)==abs(bb),f=0;break));s=s+f);s
m=0;n=0;while(1,n=n+1;res=0;for(l=1,n,rr=r(l,n);if(rr>res,res=rr));for(k=1,n-1,rr=r(n,k);if(rr>res,res=rr));if(res>m,m=res;print(n,","))) }

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

2, 0, 121, 2299, 6061, 43681, 66671, 33659659, 187891, 1266749, 8067191, 639533521, 2066801

Offset: 0

Seiichi Manyama, Jul 02 2024

a(n) is the smallest nonnegative k such that A374276(k) = n.
a(14) = 36735721.
a(15) = 153276629.
a(16) = 7703531.
a(18) = 39269219.
a(20) = 250082921.
a(24) = 84738841.
a(32) = 454508329.

Cf. A000446, A198799.

Cf. A031363, A374091, A374095, A374276.

A374276 Number of representations of n by the quadratic form x^2 + 3xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2

Offset: 0

Seiichi Manyama, Jul 02 2024

nonn

			121 = 0^2 + 3*0*11 + 11^2 = 3^2 + 3*3*7 + 7^2. So a(121) = 2.

Cf. A031363, A088534, A374093, A374275.

a[n_]:=Module[{m=Floor[Sqrt[n]]},Sum[Sum[Boole[i^2+3i*j+j^2==n],{j,i,m}],{i,0,m}]]; Array[a,122,0] (* Stefano Spezia, Jul 02 2024 *)

a(n) = my(m=sqrtint(n)); sum(i=0, m, sum(j=i, m, i^2+3*i*j+j^2==n));

a(A031363(n)) > 0.

cp's OEIS Frontend

A238240 Positive integers n such that x^2 - 20xy + y^2 + n = 0 has integer solutions.

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A031364 Number of coincidence site modules of index 10n+1 in an icosahedral module.

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A316621 Numbers of the form x^2 + 9*x*y + y^2, 0 <= x <= y.

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PARI

A336403 Multiplicative closure of A045468: numbers which are the product of zero or more primes which are 1 or 4 mod 5.

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Extensions

A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -43.

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PARI

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A238245 Positive integers n such that x^2 - 22xy + y^2 + n = 0 has integer solutions.

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A240469 Values k where the maximum number of distinct rational solutions to x^2 - Dy^2 = t, 0 < D <= k, 0 < t <= k, achieves a new record.

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Author

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Examples

Crossrefs

Programs

PARI

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3*x*y + y^2 with 0 <= x <= y.

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A316621 Numbers of the form x^2 + 9xy + y^2, 0 <= x <= y.

A035233 Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -43.

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

A374276 Number of representations of n by the quadratic form x^2 + 3xy + y^2 with 0 <= x <= y.