A094379 -id:A094379 - cp's OEIS frontend

A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe, Apr 08 2004
Note that n+1 must be prime for all n in this sequence. - T. D. Noe, Apr 28 2004
Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
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.
M. Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0A006093.
Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.
Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]

Corrected by R. H. Hardin

A066955 Number of unordered solutions of xy + yz + z*x = n, x,y,z > 0.

Original entry on oeis.org

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

Offset: 1

Colin Mallows, Jan 26 2002

a(n) is the number of distinct rectangular cuboids each one having integer surface area 2*n and integer edge lengths x, y and z. - Felix Huber, Aug 08 2023

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

Cf. A000196, A066360, A094379.

a066955 n = length [(x,y,z) | x <- [1 .. a000196 (div n 3)],
                              y <- [x .. div n x],
                              z <- [y .. div (n - x*y) (x + y)],
                              x * y + (x + y) * z == n]
-- Reinhard Zumkeller, Mar 23 2012

a(n)=sum(i=1,n,sum(j=1,i,sum(k=1,j,if(i*j+j*k+k*i-n,0,1))))

a(A094379(n)) = n and a(m) = n for m < A094379(n). - Reinhard Zumkeller, Mar 23 2012

More terms from Benoit Cloitre, Feb 02 2003

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}]

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A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

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A066955 Number of unordered solutions of x*y + y*z + z*x = n, x,y,z > 0.

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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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A066955 Number of unordered solutions of xy + yz + z*x = n, x,y,z > 0.