cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 14, 32, 52, 61, 104, 118, 133, 146, 163, 202, 242, 246, 266, 314, 342, 404, 437, 467, 472, 504, 542, 547, 577, 619, 625, 714, 757, 801, 807, 853, 907, 957, 1015, 1022, 1082, 1139, 1145, 1265, 1278, 1335, 1414, 1475
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=61: [A070080(61), A070081(61), A070082(61)]=[5=5<9], A070084(69)=gcd(5,5,9)=1, A070085(61)=5^2+5^2-9^2=25+25-81=-31<0.

Links

Crossrefs

Cf. A070107, A070110, A070127, A070115.

A070091 Number of isosceles integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 4, 2, 5, 2, 2, 2, 6, 2, 5, 3, 5, 3, 7, 2, 8, 4, 4, 4, 6, 3, 9, 4, 6, 4, 10, 4, 11, 5, 6, 5, 12, 4, 10, 5, 8, 6, 13, 4, 10, 6, 8, 7, 15, 4, 15, 7, 10, 8, 12, 6, 17, 8, 10, 6, 18, 6, 18, 9, 10, 9, 14, 6, 20, 8, 13
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n) = A051493(n) - A005044(n-6).

Examples

			For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: four are isosceles: [1<7=7], [3<6=6], [4=4<7] and [5=5=5], but GCD(3,6,6)>1 and GCD(5,5,5)>1, therefore a(15)=2.

Links

Crossrefs

Cf. A070080, A070081, A070082, A059169, A070099, A070107, A070084, A070116.

Programs

  • Mathematica
    m = 81 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &] ;
a[n_] := Count[triangles, t_ /; Total[t] == n && Length[Union[t]] < 3 && GCD @@ t == 1];
Table[a[n], {n, 1, m}] (* Jean-François Alcover, Oct 05 2021 *)

A070137 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a right integer triangle with relatively prime side lengths.

Original entry on oeis.org

17, 212, 493, 1297, 2574, 4298, 5251, 14414, 16365, 21231, 26125, 39056, 42597, 55042, 63770, 75052, 91121, 97256, 124355, 164640, 200999, 213083, 253721, 275999, 367997, 384154, 415778, 478343, 511633, 518370, 606417, 665040, 689356, 755435, 846571
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Right integer triangles have integer areas: see A070143.

Examples

			493 is a term: [A070080(493), A070081(493), A070082(493)]=[8,15,17], A070084(493)=gcd(8,15,17)=1, A070085(493)=8^2+15^2-17^2=64+225-289=0.

Links

Crossrefs

Cf. A070109, A070136.

Programs

  • Mathematica
    m = 500 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; GCD[a, b, c] == 1 && a^2 + b^2 - c^2 == 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Extensions

More terms from Jean-François Alcover, Oct 04 2021

A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 4, 2, 2, 2, 5, 1, 4, 2, 4, 3, 6, 2, 6, 3, 4, 3, 5, 3, 8, 3, 4, 3, 8, 3, 9, 5, 5, 4, 10, 3, 9, 4, 6, 5, 11, 4, 8, 5, 7, 6, 12, 3, 13, 6, 8, 7, 9, 4, 14, 7, 8, 5, 15, 5, 15, 7, 9, 8, 13, 6, 16, 6, 11, 8, 17, 5, 13
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070093, A059169, A051493, A070091, A070094, A070098, A070084, A070125.

A070107 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 1, 3, 2, 1, 1, 1, 0, 4, 2, 2, 2, 4, 1, 3, 2, 3, 2, 4, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A059169, A051493, A070091, A070102, A070106, A070084, A070134.

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 119, 122, 123, 125, 126
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			36 is a term [A070080(36), A070081(36), A070082(36)]=[3<6<7], A070084(36)=gcd(3,6,7)=1.

Links

Crossrefs

Cf. A005044, A070122, A070131, A070112, A070110.

Programs

  • Mathematica
    m = 50 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070116 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 19, 22, 23, 27, 28, 32, 35, 39, 40, 43, 46, 47, 51, 52, 55, 58, 61, 63, 64, 65, 72, 73, 81, 88, 94, 95, 98, 103, 104, 107, 108, 109, 118, 121, 124, 133, 135, 136, 140, 146, 150, 151, 159, 163, 166
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=15: [A070080(15), A070081(15), A070082(15)]=[3<4=4], A070084(15)=gcd(3,4,4)=1.

Links

Crossrefs

Cf. A070091, A070125, A070134, A070115, A070110.

A070119 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an acute integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 12, 15, 16, 19, 22, 23, 27, 28, 33, 35, 39, 40, 43, 45, 46, 47, 51, 53, 55, 58, 60, 63, 64, 65, 70, 72, 73, 81, 83, 88, 90, 92, 94, 95, 98, 103, 106, 107, 108, 109, 114, 119, 121, 124, 132, 134, 135, 136, 140, 142, 148
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(15)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070084(33)=gcd(4,5,6)=1, A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Links

Crossrefs

Cf. A070094, A070122, A070125, A070118, A070110.

A070128 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 8, 13, 14, 20, 21, 25, 29, 30, 32, 36, 37, 41, 42, 44, 49, 52, 56, 57, 59, 61, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 104, 105, 110, 111, 113, 115, 118, 122, 123, 125, 126, 127, 128, 130, 131, 133
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(9)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070084(30)=gcd(3,5,7)=1, A070085(30)=3^2+5^2-7^2=9+25-49=-15>0.

Links

Crossrefs

Cf. A070102, A070131, A070134, A070110, A070127.

A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 1, 4, 3, 6, 2, 7, 4, 8, 4, 8, 6, 10, 6, 12, 8, 14, 8, 16, 11, 18, 11, 17, 14, 21, 12, 25, 18, 25, 15, 30, 19, 32, 20, 32, 25, 38, 23, 40, 28, 41, 28, 47, 31, 51, 34, 46, 40, 55, 35, 61, 44, 58, 41, 68
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A051493, A005044, A070102, A024156, A070084, A070131.

Programs

  • Maple
    f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  for a from 1 to n/3 do
    if n::even then bmin:= max(a+1,n/2-a+1) else bmin:= max(a+1,(n+1)/2-a) fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> igcd(a,b,n-a-b) = 1, [$bmin .. bmax]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024
Previous Showing 11-20 of 21 results. Next

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