A316841 -id:A316841 - cp's OEIS frontend

A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.

0, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 3, 4, 1, 1, 4, 1, 3, 3, 3, 1, 4, 1, 3, 1, 3, 1, 8, 1, 1, 3, 3, 4, 4, 1, 3, 3, 3, 1, 7, 1, 3, 4, 3, 1, 4, 1, 3, 3, 3, 1, 4, 3, 3, 3, 3, 1, 8, 1, 3, 3, 1, 3, 7, 1, 3, 3, 7, 1, 4, 1, 3, 4, 3, 4, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3

Offset: 1

Felix Huber, Dec 27 2024

nonn

A prime factor can be used for several sides.

A nondegenerate triangle is a triangle whose sides (u, v, w) are such that u + v > w, v + w > u and u + w > v.

			a(10) = 3 because there are the 3 distinct nondegenerate triangles (2, 2, 2), (2, 5, 5), (5, 5, 5) whose sides are prime factors of 10. Since 2 + 2 < 5, (2, 2, 5) is not a triangle.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Triangle Inequality

Cf. A000040, A000961, A001221, A007947, A070088, A306678, A316841, A316842, A366398, A378820, A379033.

A378819:=proc(n)
   local a,i,j,k,L;
   L:=NumberTheory:-PrimeFactors(n);
   a:=0;
   for i to nops(L) do
      for j from i to nops(L) do
         for k from j to nops(L) while L[k]A378819(n),n=1..88);

a(n) = a(A007947(n)).

a(p^k) = 1 for prime powers p^k (p prime, k >= 1).

A378820 a(n) is the number of distinct nondegenerate triangles whose sides are divisors of n.

Original entry on oeis.org

1, 3, 3, 6, 3, 11, 3, 10, 6, 10, 3, 26, 3, 10, 11, 15, 3, 23, 3, 23, 10, 10, 3, 46, 6, 10, 10, 22, 3, 45, 3, 21, 10, 10, 11, 57, 3, 10, 10, 43, 3, 41, 3, 21, 24, 10, 3, 70, 6, 21, 10, 21, 3, 39, 10, 42, 10, 10, 3, 114, 3, 10, 23, 28, 10, 39, 3, 21, 10, 42, 3, 108

Offset: 1

Felix Huber, Dec 27 2024

nonn

A divisor can be used for several sides.

A nondegenerate triangle is a triangle whose sides (u, v, w) are such that u + v > w, v + w > u and u + w > v.

			a(4) = 6 because there are the 6 distinct nondegenerate triangles (1, 1, 1), (1, 2, 2), (1, 4, 4), (2, 2, 2), (2, 4, 4), (4, 4, 4) whose sides are divisors of 4. The triples (1, 1, 2), (1, 1, 4), (1, 2, 4), (2, 2, 4) are not sides of (nondegenerate) triangles.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Triangle Inequality

Cf. A000005, A316841, A316842, A378819.

A378820:=proc(n)
   local a,i,j,k,L;
   L:=NumberTheory:-Divisors(n);
   a:=0;
   for i to nops(L) do
      for j from i to nops(L) do
         for k from j to nops(L) while L[k]A378820(n),n=1..72);

a(p) = 3 for prime p.

A331012 16 * squared area of triangles with integer sides i <= j <= k, such that more triples of sides produce the same area as for any smaller area.

Original entry on oeis.org

3, 63, 1575, 5760, 24255, 51975, 80640, 172800, 322560, 403200, 1209600, 2822400, 4435200, 8870400, 15523200, 17740800, 53222400, 125798400, 146764800

Offset: 1

Hugo Pfoertner, Jan 07 2020

The corresponding record counts of triangles are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 26, 30, 34, 35, ... .

Conjectured further terms are 226195200 (42 representations), 230630400 (52 representations), 372556800 (55 representations).

			See A331011.

Cf. A316841, A316853, A317182, A331011.

A367196 Lexicographically earliest sequence such that for any distinct j, k, m that are the side lengths of a triangle, a(j), a(k), and a(m) are not the side lengths of a triangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 5, 1, 8, 13, 21, 2, 34, 55, 89, 1, 144, 233, 4, 377, 610, 987, 1597, 1, 17, 2584, 4181, 6765, 10946, 17711, 3, 72, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 1, 7, 305, 832040, 1346269, 2178309, 3524578, 41, 5702887, 1292, 9227465

Offset: 1

Neal Gersh Tolunsky, Nov 09 2023

nonn

In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z. The empty triangle (or line) is not counted, which means that x + y cannot be equal to z. In practice, if we have two side lengths x and y, we can find their sum s and their difference d, which tells us that side z must fall in the range d < z < s to form a triangle.
For n>0, A002620(n+1) gives the number of combinations of three indices whose corresponding terms cannot be the side lengths of a triangle in this sequence.
It appears that the local maxima are the Fibonacci numbers A000045 (except for 1s).
The second-largest values in the log graph, falling roughly on a line, appear to be A001076 (half of the even Fibonacci numbers).
Generalizing the sequence to prohibit the side lengths of any n-gon at distinct n-gonal indices gives A011782.

			a(3)=1 because the indices 1,2,3 could not be the side lengths of a triangle, so there is no restriction and the smallest number is chosen.
a(7) cannot be 1 because a(3)=1, a(5)=1, and a(7)=1 could be the side lengths of a triangle at indices which are also side lengths of a triangle.
a(7) cannot be 2 because a(4)=2, a(6)=3, and a(7)=2 are side lengths of a triangle at indices that forbid it.
a(7) cannot be 3 because a(5)=1, a(6)=3, and a(7)=3 also make a triangle at indices that forbid it.
a(7) cannot be 4 because a(4)=2, a(6)=3 and a(7)=4 form a triangle at unsuitable indices.
a(7) can be 5 without contradiction, so a(7)=5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..2000
Samuel Harkness, MATLAB program

Cf. A000045, A001076.
Cf. A229037, A276204, A364057, A361933.
Cf. A316841, A070080 (triangle side lengths).

MATLAB
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a(11)-a(50) from Samuel Harkness, Nov 13 2023

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

Original entry on oeis.org

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 120, 131, 144, 156, 168, 182, 196, 210, 225, 239, 256, 270, 288, 306, 321, 342, 361, 380, 399, 420, 441, 460, 484, 505, 527, 552, 576, 599, 623, 649, 673, 702, 729, 752, 781, 808, 840, 870, 900

Offset: 1

Hugo Pfoertner, Apr 16 2024

nonn

Cf. A316841, A371973, A371345.

See the formula section for the relationships with A002620, A173196, A316843, A316853.

A2(a,b,c) = {my(s=(a+b+c)/2);s*(s-a)*(s-b)*(s-c)};
a371972(n) = {my (A=List()); for(s2=1,n, for(s3=1,s2, if(s2+s3>n, listput(A, A2(n,s2,s3))))); #Set(A)};

a(n) <= A002620(n+1), with equality for n <= 20.

a(n) = |{A316853(k) : A316843(k) = n}| = |{A316853(k) : A173196(n) < k <= A173196(n+1)}|. - Peter Munn, Jul 30 2025

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A378819 a(n) is the number of distinct nondegenerate triangles whose sides are prime factors of n.

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A378820 a(n) is the number of distinct nondegenerate triangles whose sides are divisors of n.

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A331012 16 * squared area of triangles with integer sides i <= j <= k, such that more triples of sides produce the same area as for any smaller area.

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A367196 Lexicographically earliest sequence such that for any distinct j, k, m that are the side lengths of a triangle, a(j), a(k), and a(m) are not the side lengths of a triangle.

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A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

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Programs

PARI

Formula