A000926 -id:A000926 - cp's OEIS frontend

A232551 Number of distinct primitive quadratic forms of discriminant -4n that exist such that every prime p for which -n is a quadratic residue mod p can be represented by one of them.

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

Offset: 1

V. Raman, Nov 26 2013

This is closely related to the class number problem.
A quadratic form is primitive if the GCD of the coefficients is 1. For example, the quadratic form 2*x^2+4*y^2 is not primitive.
Two quadratic forms f(x,y) = a*x^2+b*x*y+c*y^2 and g(x,y) = p*x^2+q*x*y+r*y^2 are distinct (or inequivalent) if and only if one cannot be obtained by a linear transformation (of the variables x, y) from the other. For example, the three quadratic forms u(x,y) = 3*x^2+2*x*y+3*y^2, v(x,y) = 3*x^2+4*x*y+4*y^2 and w(x,y) = 3*x^2+10*x*y+11*y^2 are equivalent because v(x,y) = u(x+y,-y) and w(x,y) = v(x+y,y). Also, w(x,y) = u(x+2*y,-y). Similarly, the two quadratic forms s(x,y) = 8*x^2+9*y^2 and t(x,y) = 17*x^2+50*x*y+41*y^2 are equivalent because t(x,y) = s(x+2*y,x+y).
The quadratic form x^2+n*y^2 is one such form and the only such form if n = 1, 2, 3, 4, 7.
a(n) = 1 if and only if n = 1, 2, 3, 4, 7.
If n is a squarefree convenient number (A000926), a(n) represents the class number of the ring Z[sqrt(n)] if n == 1 (mod 4) or if n == 2 (mod 4) and the class number of the ring Z[(1+sqrt(n))/2] if n == 3 (mod 4) and this class number is a power of 2.
Any prime p such that -n is a quadratic residue mod p can be represented by exactly one of the a(n) distinct primitive quadratic forms of discriminant = -4n in at most four different ways (if n >= 2) or in at most eight different ways (if n = 1).
If n is a prime congruent to 3 (mod 4), then a(n) = A232550(n).
If p is a prime, p^2 does not divide n, and p > 2 if n == 3 (mod 8), then there is a multiple of p in which p is raised to an odd power which can be written in the form x^2+n*y^2 if and only if -n is a quadratic residue mod p.
The product of two numbers (prime or composite, same or different) which can be represented by the same quadratic form of discriminant = -4n can be written in the form x^2+n*y^2, as the following identity shows:
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*Z+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(a*d-b*c)^2.
(X*a^2+Y*a*b+Z*b^2)*(X*c^2+Y*c*d+Z*d^2) = (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2))^2 + ((X*Z)-(Y^2/4))*(b*d*(Y/X)+a*d+b*c)^2.
Note that for the latter equation, (a*c*X+b*d*((Y^2/(2*X))-Z)+a*d*(Y/2)+b*c*(Y/2)) and (b*d*(Y/X)+a*d+b*c) need not always be integers. If they are both integers, then it will be a second representation of the product of (X*a^2+Y*a*b+Z*b^2) and (X*c^2+Y*c*d+Z*d^2) in the form x^2+((X*Z)-(Y^2/4))*y^2.
This sequence is the same as taking every fourth number in A107628. - T. D. Noe, Jan 02 2014

			If n = 1, 2, 3, 4 or 7, then the only such available quadratic form is x^2+n*y^2.
For n = 5, every prime that is congruent to {1, 2, 3, 5, 7, 9} mod 20 can be represented by either of the two distinct primitive quadratic forms of discriminant = -20: x^2+5*y^2 or 2*x^2+2*x*y+3*y^2.
For n = 6, every prime that is congruent to {1, 2, 3, 5, 7, 11} mod 24 can be represented by either of the two distinct primitive quadratic forms of discriminant = -24: x^2+6*y^2 or 2*x^2+3*y^2.
For n = 10, every prime that is congruent to {1, 2, 5, 7, 9, 11, 13, 19, 23, 37} mod 40 can be represented by either of the two distinct primitive quadratic forms of discriminant = -40: x^2+10*y^2 or 2*x^2+5*y^2.

V. Raman, Illustration of above mentioned quadratic forms for n = 1..1000

Cf. A000003, A000926, A232529, A232530, A232550 (Number of distinct primitive quadratic forms of discriminant = -4*n needed to generate all primes p for which p is a quadratic residue (mod 4*n) or p-n is a quadratic residue (mod 4*n)).

A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 2, 3, 4, 4, 3, 3, 3, 5, 4, 2, 4, 6, 3, 4, 5, 4, 4, 4, 4, 6, 4, 3, 6, 7, 2, 4, 6, 6, 5, 4, 3, 7, 6, 3, 6, 8, 4, 5, 6, 5, 4, 6, 6, 9, 4, 2, 7, 8, 4, 5, 8, 7, 6, 6, 3, 8, 6, 4, 8, 9, 3, 6, 8, 7, 6, 4, 6, 11, 7, 3, 7, 10, 4, 6, 8, 6, 7

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe, Jun 14 2006

a(n) is also the total number of distinct quadratic forms of discriminant -4n. A232551 counts only the primitive quadratic forms of discriminant -4n (those with all coefficients pairwise coprime) and A234287 includes those by which some prime can be represented (those with all coefficients pairwise coprime or coefficient of x^2 is prime or coefficient of y^2 is prime). This sequence includes all quadratic forms like 2x^2 + 2xy + 4y^2 and 2x^2 + 4y^2 which are non-primitive and those like 4x^2 + 2xy + 4y^2 and 4x^2 + 4xy + 4y^2 by which no prime can be represented (those with no restrictions). - V. Raman, Dec 24 2013

			a(12)=4 because of (0,1,12), (0,2,6), (0,3,4), (2,2,2).
a(20)=5 because of (0,1,20), (0,2,10), (0,4,5), (1,2,6), (2,2,4).

T. D. Noe, Table of n, a(n) for n = 1..10000
V. Raman, Examples of these distinct quadratic forms for n = 1..100

Cf. A000003, A000926, A067751, A067753, A067754, A232550, A232551, A234287.

Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt++ ], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected, extended and edited by John W. Layman, Dec 03 2004

A094376 Least number having exactly n representations as ab+ac+bc with 0 < a < b < c.

Original entry on oeis.org

1, 11, 23, 41, 47, 59, 71, 116, 119, 131, 164, 425, 191, 236, 239, 446, 335, 419, 311, 404, 431, 584, 647, 524, 479, 1019, 831, 776, 671, 944, 719, 1076, 839, 1004, 959, 1889, 1196, 2099, 1271, 1856, 1151, 1931, 1391, 1676, 1319, 1616, 1751, 3275, 1511

Offset: 0

T. D. Noe and Robert G. Wilson v, Apr 28 2004

nonn

Note that the Mathematica program computes A094376, A094377 and A094378, but outputs only this sequence.

			a(2) = 23 because 23 is the least number with 2 representations: (a,b,c) = (1,2,7) and (1,3,5).

See A025052

Robert Israel, Table of n, a(n) for n = 0..826

Cf. A000926 (n having no representations), A093669 (n having one representation), A025052, A094377, A094378.

f:= proc(n) local a, t, s;
  t:= 0;
  for a from 1 to floor(sqrt(n/3)) do
    t:= t + nops(select(s -> s > 2*a and n+a^2 > s^2, numtheory:-divisors(n+a^2)))
  od;
  t
end proc:
N:= 200: # for a(0)..a(N)
V:= Array(0..N): count:= 0:
for n from 1 while count < N+1 do
   v:= f(n); if v <= N and V[v] = 0 then
      count:= count+1; V[v]:= n; fi
od:
seq(V[i],i=0..N); # Robert Israel, May 05 2021

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 1]], {i, cntMax+1}]

A094377 Greatest number having exactly n representations as ab+ac+bc with 0 < a < b < c.

Original entry on oeis.org

1848, 193, 1012, 862, 3040, 2062, 4048, 3217, 7392, 4162, 7837, 8002, 12397, 13297, 14722, 16417, 21253, 21058, 30493, 27358, 34357, 34318, 47338, 40177, 50317, 39502, 61462, 62302, 73117, 83218, 106177, 67138, 92698, 102958, 134773, 111577, 112942, 121522, 104938, 96958, 151237, 166798, 150382, 139393, 190513, 129838

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Numbers up to 250,000 were checked. Note that the Mathematica program computes A094376, A094377 and A094378, but outputs only this sequence.

			a(1) = 193 because 193 is the largest number with a unique representation: (a,b,c) = (4,7,15).

See A025052.

Cf. A000926 (n having no representations), A093669 (n having one representation), A094376, A094378.

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 2]], {i, cntMax+1}]

More terms (using limit 10^6) from Joerg Arndt, Oct 01 2017

A033266 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 2.

Original entry on oeis.org

5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 22, 25, 28, 37, 58

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 60.
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.

A subsequence of A000926.

Cf. A033267, A033268, A033269.

ok(n)={my(u=quadclassunit(-4*n).cyc); #u==1 && !select(t->t<>2, u)} \\ Andrew Howroyd, Jun 09 2018

A033267 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 4.

Original entry on oeis.org

21, 24, 30, 33, 40, 42, 45, 48, 57, 60, 70, 72, 78, 85, 88, 93, 102, 112, 130, 133, 177, 190, 232, 253

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 60.
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.

A subsequence of A000926.

Cf. A033266, A033268, A033269.

ok(n)={my(u=quadclassunit(-4*n).cyc); #u==2 && !select(t->t<>2, u)} \\ Andrew Howroyd, Jun 09 2018

A033268 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 8.

Original entry on oeis.org

105, 120, 165, 168, 210, 240, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 60.
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.

A subsequence of A000926.

Cf. A033266, A033267, A033269.

ok(n)={my(u=quadclassunit(-4*n).cyc); #u==3 && !select(t->t<>2, u)} \\ Andrew Howroyd, Jun 09 2018

A033269 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 16.

Original entry on oeis.org

840, 1320, 1365, 1848

Offset: 1

N. J. A. Sloane

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 60.
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.

A subsequence of A000926.

Cf. A033266, A033267, A033268.

ok(n)={my(u=quadclassunit(-4*n).cyc); #u==4 && !select(t->t<>2, u)} \\ Andrew Howroyd, Jun 09 2018

A094378 Number of numbers having exactly n representations as ab+ac+bc with 0 < a < b < c.

Original entry on oeis.org

65, 23, 91, 40, 197, 39, 195, 56, 298, 87, 217, 60, 512, 97, 327, 77, 562, 125, 433, 88, 712, 125, 484, 115, 924, 121

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Numbers up to 250,000 were checked. Note that there seem to be many more numbers having an even number of representations. Note that the Mathematica program computes A094376, A094377 and A094378, but outputs only this sequence.

			a(1) = 23 because there are 23 numbers (A093669) with unique representations.

See A025052

Cf. A000926 (n having no representations), A093669 (n having one representation), A094376, A094377.

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>cntMax, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt<=cntMax, If[nSol[[cnt+1, 1]]==0, nSol[[cnt+1, 1]]=n]; nSol[[cnt+1, 2]]=n; nSol[[cnt+1, 3]]++;], {n, 10000}]; Table[nSol[[i, 3]], {i, cntMax+1}]

A026501 a(n) = least positive integer > a(n-1) and not a(j)a(k) + a(k)a(i) + a(i)*a(j) for 1<=i

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 100, 102, 105, 112, 120, 130, 133, 145, 148, 165, 168, 177, 190, 210, 217, 221, 232, 240, 253, 254, 262, 263, 267, 273, 277, 280

Offset: 1

Clark Kimberling

nonn

Unlike A000926, this sequence is infinite. The first term not in A000926 is a(37) = 100. - Ivan Neretin, Jul 29 2015

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A000926.

N:= 1000: # to get all terms <= N
Allowed:= {$1..N}:
for count from 1 while Allowed <> {} do
  a:= min(Allowed);
  A[count]:= a;
  Allowed:= Allowed minus{a, seq(seq(A[i]*A[j]+(A[i]+A[j])*a,j=1..i-1),i=1..count-1)};
od:
seq(A[i],i=1..count-1); # Robert Israel, Aug 11 2015

oka(va, nv) = {for (i=1, nv, for (j=i+1, nv, for (k=j+1, nv, if (va[nv] == va[j]*va[k] + va[k]*va[i] + va[i]*va[j], return (0));););); return (1);}
finda(va) = {na = vecmax(va) + 1; va = concat(va, na); while(! oka(va, #va), va[#va] = na++); na;}
lista(nn) = {va = [1]; print1(1, ", "); for (n=1, nn, na = finda(va); print1(na, ", "); va = concat(va, na);); va;} \\ Michel Marcus, Aug 10 2015

More terms from Jud McCranie

cp's OEIS Frontend

A232551 Number of distinct primitive quadratic forms of discriminant -4n that exist such that every prime p for which -n is a quadratic residue mod p can be represented by one of them.

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A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers.

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Mathematica

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A094376 Least number having exactly n representations as ab+ac+bc with 0 < a < b < c.

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A094377 Greatest number having exactly n representations as ab+ac+bc with 0 < a < b < c.

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Mathematica

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A033266 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 2.

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PARI

A033267 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 4.

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PARI

A033268 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 8.

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PARI

A033269 Numbers n such that every genus of binary quadratic forms of discriminant -4n consists of a single class and the class number h(-4n) = 16.

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PARI

A094378 Number of numbers having exactly n representations as ab+ac+bc with 0 < a < b < c.

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A026501 a(n) = least positive integer > a(n-1) and not a(j)a(k) + a(k)a(i) + a(i)*a(j) for 1<=i