A070114 -id:A070114 - cp's OEIS frontend

A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 50, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 87, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 116, 119, 122

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(17)=50: [A070080(50), A070081(50), A070082(50)]=[4<6<8].

Jean-François Alcover, Table of n, a(n) for n = 1..920
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070113, A070114, A070121, A070122, A070123, A070130, A070131, A070132.

m = 55 (* 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] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070111 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with prime sides.

Original entry on oeis.org

3, 5, 6, 9, 14, 16, 22, 30, 34, 35, 43, 46, 63, 84, 101, 109, 124, 133, 153, 159, 163, 170, 189, 193, 201, 234, 240, 286, 297, 328, 334, 350, 352, 382, 392, 410, 444, 450, 454, 472, 478, 479, 515, 519, 527, 542, 544, 597, 603, 621, 629, 688, 708, 714, 771, 777, 795, 799, 811, 817, 868, 878, 900, 907, 911

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			35 is a term: [A070080(35), A070081(35), A070082(35)]=[2,7,7].

Reinhard Zumkeller, Integer-sided triangles

Cf. A070088, A070114, A070117, A070120, A070123, A070126, A070129, A070132, A070135.

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_} /; AllTrue[{a, b, c}, PrimeQ]] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

More terms from Jean-François Alcover, Oct 04 2021

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

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 *)

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

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A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

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A070111 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with prime sides.

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

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