A070088 -id:A070088 - cp's OEIS frontend

A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A005044, A070088, A070090, A070095, A024154, A070123.

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

Original entry on oeis.org

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).

A379033 Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a nondegenerate triangle.

Original entry on oeis.org

8, 12, 18, 27, 45, 50, 75, 98, 105, 125, 147, 175, 242, 245, 338, 343, 363, 385, 429, 507, 539, 578, 605, 637, 715, 722, 845, 847, 867, 969, 1001, 1058, 1083, 1105, 1183, 1309, 1331, 1445, 1547, 1573, 1587, 1615, 1682, 1729, 1805, 1859, 1922, 2023, 2057, 2185, 2197

Offset: 1

Felix Huber, Dec 24 2024

nonn

Subsequence of A014612 and of A145784.

Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a degenerate triangle are in A071142.

			12 = 2*2*3 is in the sequence because 2 + 2 > 3.
20 = 2*2*5 is not in the sequence because 2 + 2 < 5.
30 = 2*3*5 is not in the sequence because 2 + 3 = 5.

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

Cf. A000040, A001222, A014612, A070088, A071142, A145784, A306678, A366398, A378819.

A379033:=proc(n)
   option remember;
   local a,i,j,P;
   if n=1 then
      8
   else
      for a from procname(n-1)+1 do
         P:=[];
         if NumberTheory:-Omega(a)=3 then
            for i in ifactors(a)[2] do
               j:=0;
               while jP[3] then
               return a
            fi
         fi
      od
   fi	
end proc;
seq(A379033(n),n=1..51);

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 1

Offset: 0

Felix Huber, Jul 16 2025

0, 1, 6, 10, 28, 55 are the only triangular numbers <= 10^6 that are not perimeters of triangles whose side lengths are triangular numbers. Conjecture: There are no other triangular numbers that have this property.

			The a(31) = 2 distinct nondegenerate triangles with perimeter 31 and whose side lengths are triangular numbers are [1, 15, 15] and [6, 10, 15].

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Maple program to compute the counted triangles
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A005044, A051516, A070088, A366398, A380875, A385737, A385860, A385872.

A385736:=proc(N) # To get the first N + 1 terms.
    local p,x,y,z,i;
    p:=[];
    for z to floor((sqrt(24*N+9)-3)/6) do
        for x from z to floor((sqrt(4*N-3)-1)/2) do
            for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*N-x^2-x-z^2-z))-1)/2)) do
                p:=[op(p),z*(z+1)/2+y*(y+1)/2+x*(x+1)/2]
            od
        od
    od;
    return seq(numboccur(p,i),i=0..N)
end proc;
A385736(87);

Trivial upper bound: a(n) <= A005044(n).

a(A385737(n)) >= 1.

A308119 Sum of the smallest side lengths of all integer-sided triangles with prime side lengths and perimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 3, 2, 3, 0, 8, 2, 8, 0, 5, 0, 7, 0, 5, 2, 10, 0, 15, 2, 15, 0, 12, 0, 18, 0, 23, 2, 21, 0, 39, 2, 37, 0, 36, 0, 31, 0, 47, 2, 47, 0, 46, 0, 48, 0, 30, 0, 47, 0, 61, 2, 35, 0, 66, 2, 92, 0, 61, 0, 77, 0, 60, 0, 43, 0, 79, 2, 90

Offset: 1

Wesley Ivan Hurt, May 13 2019

nonn

Wikipedia, Integer Triangle

Cf. A010051, A070088.

Table[Sum[Sum[k (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}]

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) * k, where c is the prime characteristic (A010051).

A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.

Original entry on oeis.org

15, 21, 27, 27, 45, 45, 55, 63, 65, 81, 85, 85, 95, 99, 115, 117, 125, 125, 135, 145, 155, 171, 175, 175, 185, 189, 205, 207, 225, 235, 243, 245, 265, 275, 279, 295, 297, 315, 315, 325, 333, 335, 355, 365, 385, 387, 405, 407, 425, 451, 455, 459, 473, 475, 475

Offset: 1

Felix Huber, Dec 04 2024

nonn

			27 is twice in the sequence because there are two distinct trapezoids [p, d, q, f, h] (p and q are parallel, height h) with prime sides and height and area 27: [13, 5, 5, 5, 3], [11, 3, 7, 5, 3].

Felix Huber, Table of n, a(n) for n = 1..2633
Felix Huber, Trapezoids having prime sides and height with area A

Cf. A000040, A027750, A070088, A214602, A365049, A366398, A374594, A378148, A378149, A378150.

with(NumberTheory):
A378675:=proc(A)
   local m,p,q,i,j,d,f,h,x,y,M,T;
   if isprime(A)=false and A>1 then
      T:=[];
      M:=map(x->A/x,select(isprime,(Divisors(A)) minus {2}));
      for m in M do
         for i to pi(floor(m-1/2)) do
            q:=ithprime(i);
            p:=2*m-q;
            if isprime(p) then
               h:=A/m;
	       for x from max(4,floor((p-q+1)/2)) by 2 to (h^2-1)/2 do
	          y:=p-q-x;
	          if issqr(x^2+h^2) and issqr(y^2+h^2) then
	             d:=isqrt(y^2+h^2);
	             f:=isqrt(x^2+h^2);
	             if isprime(d) and isprime(f) then
	                T:=[op(T),A]
	             fi
	          fi
	       od
	    fi
         od
      od;
      return op(T)
   fi;
end proc;
seq(A378675(A),A=1..475);

A308165 Sum of the perimeters of all integer-sided triangles with perimeter n and prime sides.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 7, 8, 9, 0, 11, 12, 13, 0, 30, 16, 34, 0, 19, 0, 21, 0, 23, 24, 50, 0, 81, 28, 87, 0, 62, 0, 66, 0, 105, 36, 111, 0, 195, 40, 205, 0, 172, 0, 135, 0, 235, 48, 245, 0, 204, 0, 212, 0, 110, 0, 171, 0, 295, 60, 183, 0, 378, 64, 520, 0, 335, 0

Offset: 1

Wesley Ivan Hurt, May 15 2019

nonn

Wikipedia, Integer Triangle

Cf. A070088.

Table[n*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}]

a(n) = n * A070088(n).

a(n) = n * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * A010051(i) * A010051(k) * A010051(n-i-k).

A308318 Number of integer-sided triangles with perimeter n and at least one nonprime side length.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 3, 2, 4, 4, 5, 4, 6, 7, 9, 8, 11, 10, 13, 11, 14, 14, 16, 15, 18, 19, 22, 21, 25, 24, 27, 26, 30, 30, 32, 32, 35, 37, 40, 40, 45, 44, 47, 47, 51, 52, 57, 56, 61, 61, 68, 65, 72, 70, 75, 74, 82, 80, 85, 84, 88, 91, 97, 96, 103

Offset: 1

Wesley Ivan Hurt, May 19 2019

nonn

Wikipedia, Integer Triangle

Cf. A005044, A010051, A070088.

Table[Sum[Sum[(1 - (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}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * (1 - A010051(i) * A010051(k) * A010051(n-i-k)).

a(n) = A005044(n) - A070088(n).

cp's OEIS Frontend

A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

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A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

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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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A379033 Numbers that are the product of exactly three (not necessarily distinct) primes and these primes are sides of a nondegenerate triangle.

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A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

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Maple

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A308119 Sum of the smallest side lengths of all integer-sided triangles with prime side lengths and perimeter n.

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Mathematica

Formula

A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.

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Maple

A308165 Sum of the perimeters of all integer-sided triangles with perimeter n and prime sides.

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Mathematica

Formula

A308318 Number of integer-sided triangles with perimeter n and at least one nonprime side length.

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