A025052 -id:A025052 - cp's OEIS frontend

A066851 Number of ordered solutions (x,y,z) to xy + yz + zx = n with x,y,z >= 1.

0, 0, 1, 0, 3, 0, 3, 3, 3, 0, 9, 1, 3, 6, 6, 3, 9, 0, 9, 9, 6, 0, 15, 6, 3, 12, 10, 3, 15, 0, 15, 12, 6, 6, 18, 9, 3, 12, 18, 6, 21, 0, 9, 21, 9, 6, 27, 7, 9, 12, 18, 9, 15, 12, 18, 24, 6, 0, 33, 6, 15, 18, 21, 12, 18, 12, 9, 27, 18, 0, 39, 9, 9, 24, 19, 21, 18, 0, 27, 27, 18, 6, 33, 18, 6

Offset: 1

Colin Mallows, Jan 24 2002

			a(5) = 3 since there are solutions (2,1,1), (1,2,1), (1,1,2).

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

a(A025052(n))=0. Cf. A066958.

More terms from Vladeta Jovovic, Jan 25 2002

A093670 Numbers having a unique representation as ab+ac+bc, with 1 <= a <= b <= c.

Original entry on oeis.org

3, 5, 7, 8, 9, 12, 13, 14, 16, 25, 28, 34, 37, 46, 82, 142

Offset: 1

T. D. Noe, Apr 08 2004

nonn

Are there more terms?

			25 is on the list because 25 = 1*1 + 1*12 + 1*12.

See A025052

Cf. A025052 (numbers not of the form ab+ac+bc, 1<=a<=b<=c).

oneSol={}; Do[lim=Ceiling[(n-1)/2]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>=b, cnt++; If[cnt>1, Break[]]], {a, 1, lim}, {b, a, lim}]; If[cnt==1, AppendTo[oneSol, n]], {n, 10000}]; oneSol

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}]

A094380 Greatest number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

Original entry on oeis.org

462, 142, 742, 862, 2170, 2062, 3502, 2962, 5278, 5413, 7282, 8002, 11302, 11278, 14722, 13918, 18778, 21058, 30178, 30493, 30622, 34318, 47338, 31102, 44902, 43717

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Numbers up to 250,000 were checked. Note that the Mathematica program computes A094379, A094380 and A094381, but outputs only this sequence.

			a(1) = 142 because 142 is the largest number with a unique representation: (a,b,c) = (1,10,12).

See A025052

Cf. A025052 (n having no representations), A093670 (n having one representation), A094379, A094381.

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-1)/2]; 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}, {b, a, 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}]

A094381 Number of numbers having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

Original entry on oeis.org

18, 16, 61, 30, 133, 51, 119, 48, 275, 59, 217, 72, 386, 65, 292, 83, 545, 101, 332, 89, 673, 120, 453, 106, 865, 104

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 A094379, A094380 and A094381, but outputs only this sequence.

			a(1) = 16 because there are 16 numbers (A093670) with unique representations.

See A025052

Cf. A025052 (n having no representations), A093670 (n having one representation), A094379, A094380.

cntMax=10; nSol=Table[{0, 0, 0}, {cntMax+1}]; Do[lim=Ceiling[(n-1)/2]; 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}, {b, a, 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}]

A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

Original entry on oeis.org

2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

Jonathan Borwein (jborwein(AT)cecm.sfu.ca), choi(AT)cecm.sfu.ca (Stephen Choi)

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 293.
L. E. Dickson, Introduction to the theory of numbers, Dover, NY, 1929.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Experimental Mathematics, Home Page

Cf. A000926, A005843, A034169, A055745, A139826. Subsequence of A025052.

noSol = {};
Do[lim = Ceiling[(n-2)/3]; found = False; Do[If[n > a*b && Mod[n - a*b, a+b] == 0 && Quotient[n - a*b, a+b] > b, found = True; Break[]], {a, 1, lim-1}, {b, a+1, lim}]; If[!found, AppendTo[noSol, n]], {n, 1000}];
Select[noSol, EvenQ[#] && SquareFreeQ[#]&] (* Jean-François Alcover, Jul 21 2022, after T. D. Noe in A000926 *)

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

Intersection of A005843 and A139826. - Andrew Howroyd, Jun 09 2018

A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 16, 18, 21, 22, 24, 25, 30, 33, 37, 40, 42, 45, 48, 57, 58, 70, 72, 78, 85, 88, 93, 102, 105, 120, 130, 133, 165, 168, 177, 190, 210, 232, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

Offset: 1

Suggested by Eike Hertel, Hugo Pfoertner, Sep 24 2013

nonn

Conjecture: These 59 numbers are all such exceptions.
Terms are idoneal numbers (A000926) except for the six terms of A229462.
Numbers k not expressible as k = x^2 - y^2 - z^2 with x,y,z >= 1 and x > y+z.

Eike Hertel, Reguläre Dreieckspflasterungen konvexer Polygone, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/01/13, 2013 (preprint).
Eike Hertel and Christian Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete Comput Geom (2014) 51:753-759.
E. Kani, Idoneal numbers and some generalizations, Ann Sci. Math. Québec, 35 (2011), pp. 197-227.
Kival Ngaokrajang, Illustration of initial terms

Cf. A000926 (idoneal numbers), A229462 (idoneal numbers not in this sequence), A229757 (hexagon exception numbers), A025052 (numbers not of form a*b+b*c+c*a).

A055745 Squarefree numbers not of form ab + bc + ca for 1 <= a <= b <= c (probably the list is complete).

Original entry on oeis.org

1, 2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

N. J. A. Sloane

Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.

Cf. A034168, A025052, A034169.

solQ[n_, x_] := Reduce[1 <= y <= z && n == x*y + y*z + z*x, {y, z}, Integers] =!= False; solQ[n_] := Catch[xm = Ceiling[(n-1)/2]; For[x = 1, x <= xm, x++, Which[ solQ[n, x] === True, Throw[True], x == xm, Throw[False]]]] ; solQ[1] = False; Reap[ Do[ If[ SquareFreeQ[n], If[! solQ[n] , Print[n]; Sow[n]]], {n, 1, 500}]][[2, 1]] (* Jean-François Alcover, Jun 15 2012 *)

A380806 Numbers not of form a(b+1) + b(c+1) + c*(a+1) for 1<=a<=b<=c.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 16, 19, 20, 26, 31, 34, 40, 44, 55, 76, 80, 94, 124, 160, 170, 220, 271

Offset: 1

Seiichi Manyama, Feb 04 2025

Subsequence of A380807.

Cf. A025052.

A101902 Numbers n that are not of the form ab+bc+cd+de+ea with 1<=a<=b<=c<=d<=e.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 14, 18, 30, 38, 42, 62

Offset: 1

T. D. Noe, Dec 20 2004

Conjecture that this sequence is complete. Note that, except for n=1 and n=2, n-1 is prime. The case of the 3-term binary form is treated in A025052. For the 4-term case, ab+bc+cd+da, no prime is representable because the 4 terms factor as (a+c)(b+d).
The sequence is indeed complete. Each sufficiently great number can be represented in one of the following ways:
n = 7k:     {1, 2, 5, 6, k - 6}
n = 7k + 1: {1, 1, 1, 6, k - 1}
n = 7k + 2: {1, 3, 3, 6, k - 4}
n = 7k + 3: {2, 3, 4, 5, k - 5}
n = 7k + 4: {1, 2, 2, 6, k - 2}
n = 7k + 5: {1, 2, 3, 6, k - 3}
n = 7k + 6: {1, 2, 4, 6, k - 4}
Smaller numbers can be checked individually. - Ivan Neretin, Dec 14 2016

Cf. A025052 (n not of form ab + bc + ca).

nn=100; cnt5=Table[0, {nn}]; Do[n=a*b+b*c+c*d+d*e+e*a; If[n<=nn, cnt5[[n]]++ ], {a, nn}, {b, a, nn}, {c, b, nn}, {d, c, nn}, {e, d, nn}]; Flatten[Position[cnt5, 0]]

Definition corrected by Ivan Neretin, Dec 14 2016

cp's OEIS Frontend

A066851 Number of ordered solutions (x,y,z) to xy + yz + zx = n with x,y,z >= 1.

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A093670 Numbers having a unique representation as ab+ac+bc, with 1 <= a <= b <= c.

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

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A094380 Greatest number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

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A094381 Number of numbers having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

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A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

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A229461 Numbers k such that there is no convex pentagon that can be decomposed into k pairwise congruent regular equilateral triangles.

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A055745 Squarefree numbers not of form ab + bc + ca for 1 <= a <= b <= c (probably the list is complete).

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Mathematica

A380806 Numbers not of form a*(b+1) + b*(c+1) + c*(a+1) for 1<=a<=b<=c.

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A380806 Numbers not of form a(b+1) + b(c+1) + c*(a+1) for 1<=a<=b<=c.