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 1-6 of 6 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

A070090 Number of scalene integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 3, 0, 2, 0, 1, 0, 3, 0, 4, 0, 1, 0, 6, 0, 4, 0, 5, 0, 6, 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]: three are scalene: [2<6<7], [3<5<7] and [4<5<6], but only one consists of primes, therefore a(15)=1.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070088, A005044, A070092, A070097, A070105, A070114.

Programs

  • Mathematica
    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 + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

Formula

a(n) = A070088(n) - A070092(n).
a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * A010051(i)A010051(k)%20*%20A010051(n-i-k).%20-%20_Wesley%20Ivan%20Hurt">* A010051(k) * A010051(n-i-k). - _Wesley Ivan Hurt, May 13 2019

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.

A070108 Number of integer triangles with perimeter n and 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(k)<=1 until k = 140, for k = 141 there are A005044(141)=432 integer triangles, a(141)=2 as
[37=37<67]: 37+37+67 = 141 and 2*(37^2)<67^2 and 37, 67 are primes,
[41=41<59]: 41+41+59 = 141 and 2*(41^2)<59^2 and 41, 59 are primes.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A059169, A070088, A070092, A070103, A070106, A070135.

A070117 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with prime side lengths.

Original entry on oeis.org

3, 5, 6, 9, 14, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 133, 159, 163, 170, 189, 201, 234, 286, 297, 350, 352, 382, 410, 450, 472, 478, 479, 515, 527, 542, 597, 629, 688, 708, 714, 811, 817, 868, 900, 907, 981, 1021, 1033
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=43: [A070080(43), A070081(43), A070082(43)]=[3,7,7].

Links

Crossrefs

Cf. A070092, A070126, A070135, A070111, A070115.

A308147 Sum of the perimeters of all integer-sided isosceles triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 7, 8, 9, 0, 11, 12, 13, 0, 15, 16, 34, 0, 19, 0, 21, 0, 0, 24, 50, 0, 54, 28, 58, 0, 31, 0, 66, 0, 35, 36, 74, 0, 117, 40, 123, 0, 86, 0, 135, 0, 47, 48, 147, 0, 153, 0, 106, 0, 55, 0, 171, 0, 59, 60, 122, 0, 189, 64, 260, 0, 134, 0, 276, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, May 14 2019

Keywords

Links

Crossrefs

Cf. A010051, A070092.

Programs

  • Mathematica
    Table[n*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}]

Formula

a(n) = 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]) * c(i) * c(k) * c(n-i-k), where c is the prime characteristic (A010051) and [] is the Iverson bracket.
Showing 1-6 of 6 results.

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