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 1-9 of 9 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

A070123 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with prime side lengths.

Original entry on oeis.org

240, 544, 799, 911, 1262, 1568, 2621, 2681, 2856, 3369, 3648, 4246, 5194, 5541, 6576, 6626, 6725, 7441, 7503, 7565, 7902, 7944, 8882, 8956, 9332, 9452, 9472, 9888, 9988, 10421, 10498, 10502, 11075, 11079, 11622
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			240 is a term because: [A070080(240), A070081(240), A070082(240)]=[7<11<13], A070085(240)=7^2+11^2-13^2=49+121-169=1>0.

Links

Crossrefs

Cf. A070097, A070118, A070111, A070112.

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_} /; a < b < c && AllTrue[{a, b, c}, PrimeQ] && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

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.

A070132 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with prime side lengths.

Original entry on oeis.org

30, 101, 153, 193, 328, 334, 392, 444, 454, 519, 603, 621, 771, 777, 795, 878, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1999, 2023, 2037, 2066, 2193, 2223, 2253, 2454, 2663, 2903, 3055, 3321, 3363
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=519: [A070080(519), A070081(519), A070082(519)]=[5<17<19], A070085(519)=5^2+17^2-19^2=25+289-361=-47<0.

Links

Crossrefs

Cf. A070105, A070111, A070127, A070112.

A070135 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 133, 163, 472, 542, 714, 907, 1888, 2064, 2267, 2466, 3133, 4240, 4858, 5188, 7563, 8469, 9466, 9958, 11069, 12841, 13471, 14876, 17032, 17794, 20214, 20268, 21125, 21943, 22843, 22933, 23837, 25767, 25797
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(3)=133: [A070080(133), A070081(133), A070082(133)]=[7=7<11], A070085(133)=7^2+7^2-11^2=49+49-121=-23<0.

Links

Crossrefs

Cf. A070108, A070111, A070127, A070115.

A070114 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with prime side lengths.

Original entry on oeis.org

30, 101, 153, 193, 240, 328, 334, 392, 444, 454, 519, 544, 603, 621, 771, 777, 795, 799, 878, 911, 1005, 1123, 1135, 1262, 1508, 1526, 1538, 1568, 1694, 1818, 1848, 1858, 1999, 2023, 2037, 2066, 2193, 2223, 2253, 2454
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(2)=101: [A070080(101), A070081(101), A070082(101)]=[5<7<11].

Links

Crossrefs

Cf. A070090, A070123, A070132, A070112, A070111.

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.

A070120 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute 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, 240, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 544, 597, 629, 688, 708, 799, 811, 817, 868, 900, 911, 981, 1021, 1033, 1105, 1153, 1262
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(6)=34: [A070080(34), A070081(34), A070082(34)]=[5,5,5], A070085(34)=5^2+5^2-5^2=25>0.

Links

Crossrefs

Cf. A070095, A070126, A070123, A070111, A070118.

A070129 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 30, 101, 133, 153, 163, 193, 328, 334, 392, 444, 454, 472, 519, 542, 603, 621, 714, 771, 777, 795, 878, 907, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1888, 1999, 2023, 2037, 2064, 2066, 2193
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

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

Links

Crossrefs

Cf. A070103, A070132, A070135, A070111, A070127.
Showing 1-9 of 9 results.

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