A067752 -id:A067752 - cp's OEIS frontend

A067754 Number of unordered primitive solutions to xy+xz+yz=n.

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

Offset: 1

Colin Mallows, Jan 31 2002

For n = m^2 this is the number of root Descartes quadruples (-m,b,c,d).

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

			a(9)=2 because of (0,1,9) and (1,1,4) (but not (0,3,3)).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.

Cf. A067751, A067752, A067753.

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

a(n) = A066360(n) + A007875(n). - T. D. Noe, Jun 14 2006

Corrected and extended by T. D. Noe, Jun 14 2006

A067753 Number of primitive solutions in nonnegative integers of xy+xz+y*z = n.

Original entry on oeis.org

3, 6, 7, 6, 9, 12, 9, 9, 9, 12, 15, 12, 9, 18, 18, 9, 15, 12, 15, 18, 18, 12, 21, 18, 9, 24, 15, 12, 21, 24, 21, 15, 18, 18, 30, 18, 9, 24, 30, 18, 27, 24, 15, 24, 18, 18, 33, 18, 15, 24, 30, 18, 21, 24, 30, 30, 18, 12, 39, 24, 21, 30, 30, 15, 30, 36, 15, 30, 30, 24, 45, 18, 15, 36

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 9*sqrt(n). - T. D. Noe, Jun 14 2006

			a(9)=9 because of permutations of (0,1,9) and (1,1,4) (but not (0,3,3)).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A066751, A067752, A067754.

CntFunc[s_List] := Module[{len=Length[Union[s]]}, If[len==3,6,If[len==2,3,1]]]; Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z] && GCD[x,y,z]==1, cnt=cnt+CntFunc[{x,y,z}]], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected and extended by T. D. Noe, Jun 14 2006

A067751 Number of solutions in nonnegative integers of xy+xz+y*z = n.

Original entry on oeis.org

3, 6, 7, 9, 9, 12, 9, 15, 12, 12, 15, 19, 9, 18, 18, 18, 15, 18, 15, 27, 18, 12, 21, 30, 12, 24, 22, 21, 21, 24, 21, 30, 18, 18, 30, 36, 9, 24, 30, 30, 27, 24, 15, 39, 27, 18, 33, 37, 18, 30, 30, 27, 21, 36, 30, 48, 18, 12, 39, 42, 21, 30, 39, 33, 30, 36, 15, 45, 30, 24, 45, 45, 15

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 9*sqrt(n). - T. D. Noe, Jun 14 2006

			a(3)=7 because of (0,1,3),(0,3,1),(1,0,3),(1,3,0),(3,0,1),(3,1,0),(1,1,1).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A067752, A067753, A067754.

CntFunc[s_List] := Module[{len=Length[Union[s]]}, If[len==3,6,If[len==2,3,1]]]; Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt=cnt+CntFunc[{x,y,z}]], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected and extended by T. D. Noe, Jun 14 2006

A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.

Original entry on oeis.org

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

Offset: 1

V. Raman, Dec 22 2013

nonn

This is similar to A232551, except that this includes non-primitive quadratic forms like 2x^2+2xy+4y^2 and 2x^2+4y^2 because the prime 2 can be represented by both of them. But unlike A067752, we do not include quadratic forms like 4x^2+2xy+4y^2 and 4x^2+4xy+4y^2 by which no prime can be represented.
So, when n == 3 (mod 4), this includes the additional non-primitive quadratic form 2x^2+2xy+((n+1)/2)y^2 and when p^2 divides n, where p is prime, this includes the additional non-primitive quadratic form px^2+(n/p)y^2.
If p is a prime and if p^2 does not divide n, then there exist a unique non-primitive quadratic form of discriminant = -4n by which p can be represented if and only if -n is a quadratic residue (mod p) and there exists a multiple of p which can be written in the form x^2+ny^2 in which p appears raised to an odd power, except when p = 2 and n == 3 (mod 8).

V. Raman, Examples of these distinct quadratic forms for n = 1..100

Cf. A000003, A000926, A067752, A232550, A232551.

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A067754 Number of unordered primitive solutions to xy+xz+yz=n.

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A067753 Number of primitive solutions in nonnegative integers of x*y+x*z+y*z = n.

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A067751 Number of solutions in nonnegative integers of x*y+x*z+y*z = n.

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A234287 Number of distinct quadratic forms of discriminant -4n by which some prime can be represented.

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A067753 Number of primitive solutions in nonnegative integers of xy+xz+y*z = n.

A067751 Number of solutions in nonnegative integers of xy+xz+y*z = n.