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.

Previous Showing 11-20 of 20 results.

A070100 Number of integer triangles with perimeter n and prime side lengths which are acute and isosceles.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070093, A059169, A070088, A070092, A070095, A070098, A070126.

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070093, A005044, A070088, A070090, A070095, A024154, A070123.

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.

A070140 Number of acute integer triangles with perimeter n having integral area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

a(n) = A051516(n) - A070141(n) - A024155(n).

Links

Crossrefs

Cf. A051516, A070093, A070146.

A070205 Number of acute integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

Cf. A024155, A070093, A070140, A070201, A070206.

Formula

a(n) = A070201(n) - A024155(n) - A070206(n).

A070096 Number of integer triangles with perimeter n and relatively prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 2, 4, 4, 4, 4, 5, 4, 6, 5, 8, 5, 9, 7, 8, 7, 8, 10, 10, 9, 12, 10, 12, 11, 14, 11, 15, 13, 15, 14, 18, 13, 19, 15, 21, 16, 23, 17, 24, 20, 25, 21, 28, 20, 28, 23, 28
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070093, A005044, A051493, A070094, A024154, A070084, A070122.

A307729 Sum of the smallest sides of all acute integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 2, 4, 5, 4, 6, 8, 6, 13, 11, 18, 17, 19, 23, 22, 30, 29, 32, 37, 40, 54, 49, 63, 63, 65, 73, 79, 94, 90, 98, 113, 110, 138, 127, 151, 154, 161, 181, 180, 210, 209, 216, 240, 247, 279, 269, 297, 320, 318, 348, 359, 402, 396, 401, 441, 440
Offset: 1

Views

Author

Wesley Ivan Hurt, May 15 2019

Keywords

Links

Crossrefs

Cf. A070093.

Programs

  • Mathematica
    Table[Sum[Sum[k*(1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) k.

A308117 Number of acute integer-sided triangles with perimeter n and squarefree sides.

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, May 13 2019

Keywords

Links

Crossrefs

Cf. A008683, A070093.

Programs

  • Mathematica
    Table[Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^2 (1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))) mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).

A307728 Sum of the perimeters of all acute integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 3, 0, 5, 6, 7, 8, 18, 20, 22, 24, 39, 28, 60, 48, 85, 72, 95, 100, 105, 132, 138, 144, 175, 182, 243, 224, 290, 270, 310, 320, 363, 408, 420, 432, 518, 494, 624, 560, 697, 672, 731, 792, 810, 920, 940, 960, 1078, 1100, 1224, 1196, 1325, 1404, 1430
Offset: 1

Views

Author

Wesley Ivan Hurt, May 15 2019

Keywords

Links

Crossrefs

Cf. A070093.

Programs

  • Mathematica
    Table[n*Sum[Sum[(1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = n * A070093(n).
a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))).

A308118 Number of obtuse integer-sided triangles with perimeter n and squarefree sides.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 2, 4, 3, 1, 0, 2, 1, 4, 5, 6, 5, 2, 2, 5, 3, 4, 3, 7, 4, 3, 2, 7, 4, 9, 8, 12, 10, 10, 8, 7, 5, 7, 7, 5, 4, 2, 4, 7, 7, 11, 12, 15, 12, 13, 13, 19, 12, 20, 20, 23, 18, 16, 16, 21, 18, 25, 27, 31
Offset: 1

Views

Author

Wesley Ivan Hurt, May 13 2019

Keywords

Links

Crossrefs

Cf. A008683, A070093, A308117.

Programs

  • Mathematica
    Table[Sum[Sum[MoebiusMu[i]^2*MoebiusMu[k]^2*MoebiusMu[n - k - i]^2 (1 - Sign[Floor[(i^2 + k^2)/(n - i - k)^2]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((i^2+k^2)/(n-i-k)^2))) * sign(floor((i+k)/(n-i-k+1))) mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Möbius function (A008683).
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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