A070088 -id:A070088 - cp's OEIS frontend

A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a + b + c = n.

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, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 3

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

From Peter Kagey, Jan 30 2018: (Start)
a(k) > 0 if and only if k is in A096468.
Records appear at indices 12, 36, 54, 84, 98, 162, 242, 338, 484, 578, ....
a(2k - 1) = 0 for all integers k > 0.
(End)

Peter Kagey, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051493, A051516, A070088, A070109, A070143, A096468, A239246.

Corrected by T. D. Noe, Jun 17 2004

A070142 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area.

Original entry on oeis.org

17, 39, 52, 116, 212, 252, 269, 368, 370, 372, 375, 493, 561, 587, 659, 839, 850, 862, 957, 972, 1156, 1186, 1196, 1204, 1297, 1582, 1599, 1629, 1912, 1920, 1955, 1971, 1988, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 2790

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070088, A070149, A070143, A070144, A070145, A051516, A069594, A069597, A070136, A070137, A070209.

maxPerim = 100; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2]&]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; Position[ stri, tri_ /; IntegerQ[area[tri]]] // Flatten (* Jean-François Alcover, Feb 22 2013 *)

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

A366398 a(n) is the number of distinct triangles with prime sides and whose perimeter is equal to the n-th prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 1, 3, 2, 3, 5, 4, 5, 4, 5, 3, 5, 4, 3, 6, 7, 10, 11, 10, 8, 12, 8, 11, 11, 12, 15, 12, 20, 19, 16, 21, 21, 21, 25, 19, 17, 15, 20, 20, 25, 36, 41, 38, 39, 34, 26, 25, 30, 34, 31, 27, 34, 45, 36, 33, 42, 39, 33, 45, 47, 54, 55, 48, 50, 58

Offset: 1

Felix Huber, Oct 09 2023

nonn

			For n = 13 the a(13) = 5 distinct triangles with prime sides (u, v, w) are (3, 19, 19), (5, 17, 19), (7, 17, 17), (11, 11, 19), and (11, 13, 17). They all have perimeter 41, which is the 13th prime.

Felix Huber, Table of n, a(n) for n = 1..1000

Cf. A070088.

A366398 := proc(n) local u, v, w, a; u := 1; a := 0; while 2*ithprime(u) < ithprime(n) do v := u; while 2*ithprime(v) <= ithprime(n) - ithprime(u) do if ithprime(n) < 2*ithprime(u) + 2*ithprime(v) and isprime(ithprime(n) - ithprime(u) - ithprime(v)) then a := a + 1; end if; v := v + 1; end do; u := u + 1; end do; return a; end proc; seq(A366398(n), n = 1 .. 100);

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

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 *)

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

A070103 Number of obtuse integer triangles with perimeter n and prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 4, 0, 5, 0, 4, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 0, 6, 0, 4, 0, 6, 0, 6, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=11 there are A005044(11)=4 integer triangles: [1,5,5], [2,4,5], [3,3,5] and [3,4,4]; only one of the two obtuses ([2,4,5] and [3,3,5]) consists of primes, therefore a(11)=1.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A070098, A070101, A070088, A070105, A070108, A070129.

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[(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}] (* Wesley Ivan Hurt, May 13 2019 *)

a(n) = A070093(n) - A070098(n).

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))) * A010051(i) * A010051(k) * A010051(n-i-k). - 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

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

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

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 *)

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

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

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

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

A070105 Number of integer triangles with perimeter n and prime side lengths which are obtuse and scalene.

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, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 5, 0, 5, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

a(n) = 0 if n is even. - Robert Israel, Jul 26 2024

Robert Israel, Table of n, a(n) for n = 1..10000
R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070101, A005044, A070088, A070090, A070103, A024156, A070132.

f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  if n::even then return 0 fi;
  for a from 1 to n/3 by 2 do
    if not isprime(a) then next fi;
    bmin:= max(a+1,(n+1)/2-a); if bmin::even then bmin:= bmin+1 fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> isprime(b) and isprime(n-a-b), [seq(b,b=bmin .. bmax,2)]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

cp's OEIS Frontend

A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a + b + c = n.

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A070142 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area.

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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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A366398 a(n) is the number of distinct triangles with prime sides and whose perimeter is equal to the n-th prime.

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Maple

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

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A070103 Number of obtuse integer triangles with perimeter n and prime side lengths.

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

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Mathematica

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

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Mathematica

Formula

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

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