A066955 -id:A066955 - cp's OEIS frontend

A025051 Numbers of the form jk + ki + i*j, where i,j,k >= 1.

3, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Clark Kimberling

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Uniquification of sequence A025050.

Cf. A025052 (complement), A066955.

Take[Union[Total[Times@@@Subsets[#,{2}]]&/@Tuples[Range[25],{3}]], 80] (* Harvey P. Dale, Aug 11 2011 *)

On the GRH, a(n) = n + 18 for n > 444. If there is some N > 444 with a(N) not equal to n + 19, then for n >= N, a(n) = n + 19 (and GRH is false). See Borwein & Choi. - Charles R Greathouse IV, Dec 19 2022

n is in the sequence if and only if A066955(n) > 0. - Charles R Greathouse IV, Jun 25 2024

A094379 Least number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

Original entry on oeis.org

1, 3, 11, 23, 35, 47, 59, 71, 95, 188, 119, 164, 231, 191, 215, 239, 299, 356, 335, 311, 404, 431, 591, 584, 524, 479, 551, 656, 831, 776, 671, 719, 791, 839, 1004, 1031, 959, 1244, 1196, 1439, 1271, 1151, 1931, 1847, 1391, 1319, 1811, 1784, 1616, 1511, 1799

Offset: 0

T. D. Noe, Apr 28 2004

nonn

Note that the Mathematica program computes A094379, A094380 and A094381, but outputs only this sequence.

A066955(a(n)) = n and A066955(m) = n for m < a(n). [Reinhard Zumkeller, Mar 23 2012]

			a(3) = 23 because 23 is the least number with 3 representations: (a,b,c) = (1,1,11), (1,2,7) and (1,3,5).

See A025052

Reinhard Zumkeller, Table of n, a(n) for n = 0..125

Cf. A025052 (n having no representations), A093670 (n having one representation), A094380, A094381.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a094379 = (+ 1) . fromJust . (`elemIndex` a066955_list)
-- Reinhard Zumkeller, Mar 23 2012

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, 1]], {i, cntMax+1}]

A238097 Number of monic cubic polynomials with coefficients from {1..n} and maximum coefficient equal to n, for which all three roots are integers.

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

Offset: 1

N. J. A. Sloane, Feb 22 2014

nonn

			a(11) = 2 with polynomials x^3 + 6*x^2 + 11*x + 6 = (x+1) * (x+2) * (x+3) and x^3 + 7*x^2 + 11*x + 5 = (x+1)^2 * (x+5). - _Michael Somos_, Feb 23 2014

Zak Seidov, Table of n, a(n) for n = 1..1000
Dorin Andrica and Eugen J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, Vol. 22(1),2014, 13-23.

Cf. A006218, A066955, A238096, A238098.

Table[p = Flatten[Table[{a, b, c, 1}, {a, n}, {b, n}, {c, n}], 2]; cnt = 0; Do[If[Max[p[[i]]] == n, poly = p[[i]].x^Range[0, 3]; r = Rest[FactorList[poly]]; If[Total[Transpose[r][[2]]] == 3 && Union[Coefficient[Transpose[r][[1]], x]] == {1}, Print[{n, r}]; cnt++]], {i, Length[p]}]; cnt, {n, 20}] (* T. D. Noe, Feb 22 2014 *)

{a(n) = if( n<1, 0, sum(a1=1, n, sum(a2=1, n, sum(a3=1, n, vecmax([a1, a2, a3]) == n && vecsum( factor( Pol([1, a1, a2, a3]))[, 2]) == 3))))}; /* Michael Somos, Feb 23 2014 */

Definition corrected by Giovanni Resta, Feb 22 2014

Extended by T. D. Noe, Feb 22 2014

A369951 Volumes of integer-sided cuboids in which either the surface area divides the volume or vice versa (assuming dimensionless unit of length).

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 27, 32, 36, 216, 250, 256, 288, 400, 432, 450, 486, 576, 882, 1728, 1800, 1944, 2000, 2048, 2304, 2744, 2916, 3200, 3456, 3528, 3600, 3888, 4608, 6144, 6174, 6750, 6912, 7056, 7200, 7350, 7776, 7986, 8000, 8100, 8232, 9000, 9216, 9600, 9800

Offset: 1

Felix Huber, Feb 12 2024

For n <= 9, the surface area divides the volume. The 9 triples with the edge lengths (u,v,w) are (1,1,1), (2,1,1), (2,2,1), (2,2,2), (4,4,1), (6,3,1), (3,3,3), (4,4,2), (6,3,2).
For 10 <= n <= 19 the surfaces and volumes are equal. This is sequence A230400.
For n >= 20 the volume divides the surface area.

			a(9) = 36, because V = 6*3*2 = 36 and S = 2*(6*3+3*2+6*2) = 72 and S/V = 2.
a(12) = 256, because V = 8*8*4 = 256 and S = 2*(8*8+8*4+8*4) = 256 and S=V.
a(20) = 1728, because V = 12*12*12 = 1728 and S = 6*12*12 = 864 and V/S = 2.

Cf. A230400 (subsequence), A066955.

A369951 := proc(V) local a, b, c, k; for a from ceil(V^(1/3)) to V do if V/a = floor(V/a) then for b from ceil(sqrt(V/a)) to floor(V/a) do c := V/(a*b); if c = floor(c) then k := 2*(a*b + c*b + a*c)/(a*b*c); if k = floor(k) or 1/k = floor(1/k) then return V; end if; end if; end do; end if; end do; end proc; seq(A369951(V), V = 1 .. 10000);

For 10 <= n <= 19, a(n) = A230400(n - 9).

A375785 a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 5, 7, 9, 9, 9, 13, 9, 9, 19, 15, 13, 19, 13, 23, 19, 19, 17, 29, 25, 19, 27, 23, 21, 41, 21, 31, 35, 29, 33, 45, 25, 29, 35, 51, 29, 41, 29, 45, 61, 39, 33, 61, 33, 57, 51, 45, 37, 63, 61, 51, 51, 49, 41, 97, 41, 49, 61, 63, 61, 81, 45, 67, 67

Offset: 1

Felix Huber, Sep 17 2024

nonn

a(n) is the number of unordered solutions (x, y, z) to x*y + y*z + x*z = 3*n^2 in positive integers x and y.

Conjecture: All terms are odd.

			a(6) = 5 because exactly the 5 integer-sided cuboids (2, 2, 26), (2, 5, 14), (2, 6, 12), (3, 6, 10), (6, 6, 6) have the same surface as a cube with edge length 6: 2*(2*2 + 2*26 + 2*26) = 2*(2*5 + 5*14 + 2*14) = 2*(2*6 + 6*12 + 2*12) = 2*(3*6 + 6*10 + 3*10) = 2*(6*6 + 6*6 + 6*6) = 6*6^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple programs
Eric Weisstein's World of Mathematics, Cuboid

Cf. A000578, A003167, A066955, A369951, A375580, A375786, A376074.

Maple
```
See Huber link.
```

A066360 Number of unordered solutions in positive integers of xy + xz + yz = n with gcd(x,y,z) = 1.

Original entry on oeis.org

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

Offset: 1

Colin Mallows, Dec 20 2001

These correspond to Descartes quadruples (-s, s+x+y, s+x+z, s+y+z) where s = sqrt(n), which are primitive if n is a perfect square.
Many empirical regularities are known, e.g., for n = 2^(2k) or n=2^(2k-1), (2 <= k <= 10 and even k <= 20), a(n) = 2^(k-2).
It appears that a(n) > 0 for n > 462. An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe, Jun 14 2006

			a(81) = 3 because we have the triples (x,y,z) = (1,1,40),(2,3,15),(3,6,7) (and not (3,3,12) because this is not primitive).

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

Cf. A060790, A062536 (and A007875 for xy = n).

Cf. A000196, A066955.

a066360 n = length [(x,y,z) | x <- [1 .. a000196 n],
                              y <- [x .. div n x],
                              z <- [y .. n - x*y],
                              x*y+(x+y)*z == n, gcd (gcd x y) z == 1]
-- Reinhard Zumkeller, Mar 23 2012

Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z] && GCD[x,y,z]==1, cnt++ ], {x,Sqrt[n/3]}, {y,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

A290870 a(n) is the number of ways to represent n as n = xy + yz + z*x where 0 < x < y < z.

Original entry on oeis.org

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

Offset: 1

Joerg Arndt, Aug 13 2017

nonn

a(n) = 0 if and only if n is a term of A000926.

a(n) = 1 if and only if n is a term of A093669.

			For (x, y, z) = (1, 3, 5), we have x * y + y * z + z * x = 1 * 3 + 3 * 5 + 5 * 1 = 23 and similarily for (x, y, z) = (1, 2, 7), we have x * y + y * z + z * x = 23. Those 2 triples are all for n=23, so a(23) = 2. - _David A. Corneth_, Oct 01 2017

Joerg Arndt, Table of n, a(n) for n = 1..10000

Cf. A066955 (ways to represent n as n = x*y + y*z + z*x where 0 <= x <= y <= z).
Cf. A094377 (greatest number having exactly n representations).
Cf. A094376 (indices of records).

N=10^3; V=vector(N);
{ for (x=1, N,
    for (y=x+1, N, t=x*y; if( t > N, break() );
      for (z=y+1, N,
        tt = t + y*z + z*x;  if( tt > N, break() );
        V[tt]+=1;
); ); ); }
V \\ Joerg Arndt, Oct 01 2017

a(n) = {my(res = 0);
for(x = 1, sqrtint(n\3), for(y = x + 1, (n - x^2) \ (2 * x), z = (n - x*y) / (x + y); if(z > y && z == z\1, res++))); res} \\ David A. Corneth, Oct 01 2017

For the triples (x,y,z) we have x < sqrt(n / 3), y < (n - x^2) / (2 * x), z = (n - x*y) / (x + y) which must be integer. - David A. Corneth, Oct 01 2017

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 0, 5, 1, 1, 2, 3, 3, 2, 1, 1, 3, 6, 1, 4, 2, 7, 4, 4, 0, 3, 5, 3, 4, 2, 1, 7, 2, 1, 5, 9, 3, 5, 3, 4, 1, 9, 2, 6, 3, 5, 6, 5, 4, 7, 5, 1, 5, 6, 3, 13, 7, 8, 4, 6, 0, 4, 4, 11, 5, 13, 2

Offset: 1

Felix Huber, Apr 04 2025

nonn

a(n) is the number of distinct cuboids with edge length 4*n whose surface area is half of a square.

Conjecture: a(k) = 0 iff k is an element of {2, 4, 8, 13} union A000244 union A005030.

			The a(14) = 3 partitions [x, y, z] are [1, 1, 12], [1, 4, 9] and [4, 4, 6] because 1*1 + 1*12 + 1*12 = 5^2, 1*4 + 4*9 + 1*9 = 7^2 and 4*4 + 4*6 + 4*6 = 8^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the partitions

Cf. A000244, A005030, A066955, A069905, A338939, A375512, A375576, A375580, A375731.

A382407:=proc(n)
    local a,x,y,z;
    a:=0;
    for x to n/3 do
        for y from x to (n-x)/2 do
            z:=n-x-y;
            if issqr(x*y+x*z+y*z) then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A382407(n),n=1..87);

cp's OEIS Frontend

A025051 Numbers of the form j*k + k*i + i*j, where i,j,k >= 1.

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A094379 Least number having exactly n representations as ab+ac+bc with 1 <= a <= b <= c.

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Mathematica

A238097 Number of monic cubic polynomials with coefficients from {1..n} and maximum coefficient equal to n, for which all three roots are integers.

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Mathematica

PARI

Extensions

A369951 Volumes of integer-sided cuboids in which either the surface area divides the volume or vice versa (assuming dimensionless unit of length).

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Comments

Examples

Crossrefs

Programs

Maple

Formula

A375785 a(n) is the number of distinct integer-sided cuboids having the same surface as a cube with edge length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A066360 Number of unordered solutions in positive integers of xy + xz + yz = n with gcd(x,y,z) = 1.

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Comments

Examples

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Crossrefs

Programs

Haskell

Mathematica

Extensions

A290870 a(n) is the number of ways to represent n as n = x*y + y*z + z*x where 0 < x < y < z.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that x*y + y*z + x*z is a perfect square.

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Author

A025051 Numbers of the form jk + ki + i*j, where i,j,k >= 1.

A290870 a(n) is the number of ways to represent n as n = xy + yz + z*x where 0 < x < y < z.

A382407 a(n) is the number of partitions n = x + y + z of positive integers such that xy + yz + x*z is a perfect square.