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.

Showing 1-5 of 5 results.

A070088 Number of integer-sided triangles with perimeter n and prime sides.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 0, 2, 0, 2, 0, 3, 1, 3, 0, 5, 1, 5, 0, 4, 0, 3, 0, 5, 1, 5, 0, 4, 0, 4, 0, 2, 0, 3, 0, 5, 1, 3, 0, 6, 1, 8, 0, 5, 0, 5, 0, 4, 0, 3, 0, 5, 1, 6, 0, 6, 0, 4, 0, 7, 1, 7, 0, 9, 1, 10, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

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]: two of them consist of primes, therefore a(15)=2.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070090, A070092, A070095, A070097, A070100, A070103, A070105, A070108, A070111.

Programs

  • Mathematica
    triangleQ[sides_] := With[{s = Total[sides]/2}, AllTrue[sides, # < s&]];
a[n_] := Select[IntegerPartitions[n, {3}, Select[Range[Ceiling[n/2]], PrimeQ]], triangleQ] // Length; Array[a, 90] (* Jean-François Alcover, Jul 09 2017 *)
Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

Formula

a(n) = A070090(n) + A070092(n) = A070095(n) + A070103(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * c(i) * c(k) * c(n-i-k), where c = A010051. - Wesley Ivan Hurt, May 13 2019

A070098 Number of integer triangles with perimeter n which are acute and isosceles.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Equivalently, the number of obtuse isosceles integer triangles with base n. - Charlie Marion, Jun 18 2019

Examples

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; both isosceles are also acute.

Links

Crossrefs

Cf. A070080, A070081, A070082, A059169, A070099, A070100, A070124.

Programs

  • Magma
    [Floor(k/2)-Floor(k/(2 + Sqrt(2)))-((k + 1) mod 2): k in [1..76]]; // Marius A. Burtea, Jun 21 2019

Formula

a(n) = A070093(n)-A024154(n); a(n) = A059169(n)-A070106(n).
a(n) = floor(n/2) - floor(n/(2 + sqrt(2))) - ((n + 1) mod 2). - David Pasino, Jun 27 2016
a(n) = A004526(n-1) - A183138(n). - R. J. Mathar, May 22 2019

A070126 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with prime side lengths.

Original entry on oeis.org

3, 6, 9, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 159, 170, 189, 201, 234, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 597, 629, 688, 708, 811, 817, 868, 900, 981, 1021, 1033, 1105, 1153, 1284, 1386, 1419, 1425
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=22: [A070080(22), A070081(22), A070082(22)]=[3<5=5], A070085(22)=3^2+5^2-5^2=9>0.

Links

Crossrefs

Cf. A070100, A070115, A070111, A070118.

A070092 Number of isosceles integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: four are isosceles: [1<8=8], [3<7=7], [5=5<7] and [5<6=6], but only two of them consist of primes, therefore a(17)=2.

Links

Crossrefs

Cf. A070080, A070081, A070082, A059169, A070100, A070108, A070117.

Programs

  • Mathematica
    Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (KroneckerDelta[i, k] + KroneckerDelta[i, n - i - k] - KroneckerDelta[k, n - i - k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 14 2019 *)

Formula

a(n) = A070088(n) - A070090(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * ([i = k] + [i = n-i-k] - [k = n-i-k]) * A010051(i) * A010051(k) * A010051(n-i-k), where [] is the Iverson bracket. - Wesley Ivan Hurt, May 14 2019

A070095 Number of acute integer triangles with perimeter n and prime side lengths.

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			For n=17 there are A005044(17)=8 integer triangles: [1,8,8], [2,7,8], [3,6,8], [3,7,7], [4,5,8], [4,6,7], [5,5,7] and [5,6,6]: the two consisting of primes ([3,7,7] and [5,5,7]) are also acute, therefore a(17)=2.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070088, A070093, A070097, A070100, A070103, A070120.

Programs

  • Mathematica
    Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) (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}] (* Wesley Ivan Hurt, May 13 2019 *)

Formula

a(n) = A070088(n) - A070103(n).
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))) * A010051(i) * A010051(k) * A010051(n-i-k). - Wesley Ivan Hurt, May 13 2019
Showing 1-5 of 5 results.

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