A106505 -id:A106505 - cp's OEIS frontend

A106506 Length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).

5, 7, 16, 9, 11, 33, 24, 13, 39, 15, 56, 32, 17, 51, 85, 19, 72, 57, 40, 95, 21, 120, 105, 23, 88, 69, 48, 115, 25, 161, 75, 27, 104, 175, 56, 135, 29, 208, 189, 87, 168, 145, 31, 120, 203, 93, 64, 261, 155, 33, 240, 217, 192, 279, 165, 35, 136, 231, 105, 320, 72, 37, 272

Offset: 1

Lekraj Beedassy, May 04 2005

nonn

Cf. A106499-A106505, A106410, A106420, A106430.

sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers];
Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[s^2 - r^2 /. sp[[1]]]; Sow[s^2 - r^2 /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

a(n) = A106501(n)^2 - A106500(n)^2.

Extended by Ray Chandler, May 09 2005

A229849 Consider all primitive 120-degree triangles with sides A < B < C. The sequence gives the values of B.

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 39, 45, 51, 56, 57, 63, 77, 80, 85, 88, 91, 95, 105, 112, 115, 120, 143, 145, 155, 160, 161, 165, 168, 175, 187, 192, 195, 203, 208, 209, 217, 221, 224, 231, 247, 253, 259, 261, 272, 273, 279, 280, 287, 288, 299, 301, 304, 315, 320, 323

Offset: 1

Colin Barker, Oct 06 2013

nonn

A primitive triangle is one for which the sides have no common factor.

For n>1, A106505(n) seems to give the values of A and A004611(n) seems to give the values of C.

			33 appears in the sequence because there exists a primitive 120-degree triangle with sides 7, 33 and 37.

Wikipedia, Integer triangle

Cf. A004611, A106505, A229858, A229859.

\\ Gives values of B not exceeding bmax
\\ e.g. pt120b(80) gives [5, 8, 16, 24, 33, 35, 39, 45, 51, 56, 57, 63, 77, 80]
pt120b(bmax) = {
  s=[];
  for(m=1, (bmax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, b=a);
        if(b<=bmax, s=concat(s, b))
      )
    )
  );
  vecsort(s,,8)
}

A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 33, 35, 37, 39, 39, 40, 41, 43, 45, 47, 48, 49, 51, 51, 53, 55, 56, 56, 57, 57, 59, 61, 63, 64, 65, 67, 69, 69, 71, 72, 72, 73, 75, 75, 77, 79, 80, 81, 83, 85, 85, 87, 87, 88, 88, 89, 91, 93, 93, 95, 95, 96, 97, 99

Offset: 1

Bernard Schott, Apr 10 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
Equivalently, length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

			According to inequalities between a, b, c, there exist 3 types of such triangles:
   c < a < b for the smallest side a = 5 and triple (5, 6, 4).
The first side a for which there exist two distinct triangles with B = 2C is for a = 33 with the two other types of examples,
   c < b < a with triple (33, 28, 16),
   a < c < b with triple (33, 272, 256).

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Cf. A353619 (similar, but with B = 3*C).
Cf. A343063 (triples), this sequence (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).
Cf. A106505 (sides a without repetition), A106506 (sides a sorted on perimeter).

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and  igcd(a,sqrt(d),c)=1 and abs(a-c)

a(n) = A343063(n, 1).

A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 115

Offset: 1

M. F. Hasler, Nov 22 2018

Equivalently, numbers of the form (x - y)^2*(x + y) or d^2*(2m + d), for (x, y) = (m+d, m). This shows that this consists of all squares d^2 > 0 times all numbers of the same parity and larger than d. In particular, for d=1, all odd numbers > 1, and for d=2, 4*(even numbers > 2) = 8*(any number > 1). Larger d can't yield additional terms, neither odd nor even: The sequence consists exactly of all odd numbers > 2 and multiples of 8 larger than 8.

			a(1) = 3 = 1*3 = (2 - 1)*(2^2 - 1^2). Similarly any larger odd number 2k+1 = (k+1 - k)((k+1)^2 - k^2) is in this sequence.
a(8) = 16 = 2*8 = (3 - 1)*(3^2 - 1^2). Similarly, any larger multiple of 8, 8*(1 + k) = 2*(4k + 4) = (k+2 - k)((k+2)^2 - k^2) is in this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

See A321491 for numbers of the form (x+y)(x^2+y^2).
Cf. A321501 (complement).
See A321498 for numbers that have two representations of the form (x-y)(x^2-y^2).
Cf. A106505 (conjectured to be the sequence without the 3).

is(n)={n&&fordiv(n,d, d^2*(d+2)>n && break; n%d^2&&next; bittest(n\d^2-d,0)||return(1))} \\ This uses the definition. More efficient variant below.

select( is_A321499(n)=if(bittest(n,0),n>1,n%8,0,n>8), [0..99]) \\ Defines the function is_A321499(). The select() command is just an illustration and check.

A321499_list(M)=setunion(vector(M\2-1,k,2*k+1),[2..M\8]*8) \\ list all terms up to M; more efficient than select() above.

apply( A321499(n)=if(n<8, 2*n+1, n%5!=3, (n+2)*4\5*2+1, n\5*8+8), [1..30]) \\ Defines A321499(n). The apply() command provides a check & illustration.

def A321499(n): return (n<<1)+1 if n<4 else (((n+2)<<2)//5<<1)+(n%5!=3) # Chai Wah Wu, Feb 26 2025

Asymptotic density is 5/8. Complement is A321501.
a(5k-2) = 8k for all k > 1, a(n) = floor((n+2)*4/5)*2 + 1 for all other n > 3.
a(n + 5) = a(n) + 8 for n > 3. - David A. Corneth, Nov 23 2018
O.g.f. 3*x+5*x^2+7*x^3 -x^4*(-9-2*x-2*x^2-2*x^3-x^4+8*x^5) / ( (x^4+x^3+x^2+x+1) *(x-1)^2 ). - R. J. Mathar, Nov 29 2018

A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 56, 25, 75, 15, 104, 32, 56, 29, 17, 87, 85, 119, 31, 72, 93, 64, 144, 19, 95, 133, 40, 136, 35, 105, 21, 105, 37, 111, 185, 88, 152, 176, 23, 80, 115, 161, 41, 123, 240, 48, 205, 240, 43, 25, 129, 175, 215, 88

Offset: 1

Bernard Schott, Sep 23 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the a's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions a of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, a is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with a = u^2 - v^2.
This sequence is not increasing. For example, a(2) = 7 for triangle with largest side = 13 while a(3) = 5 for triangle with largest side = 19.
Differs from A088514, the first 20 terms are the same then a(21) = 56 while A088514(21) = 25.
A229858 gives all the possible values of the smallest side a, in increasing order without repetition, but for all triples, not necessarily primitive.
All terms of A106505 are values taken by the smallest side a, in increasing order without repetition for primitive triples, but not all the lengths of this side a are present; example: 3 is not in A106505 (see comment in A229849).

			a(2) = a(5) = 7 because 2nd and 5th triple are respectively (7, 8, 13) and (7, 33, 37).

Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A002324, A050931, A088514, A106505, A229849.

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a

Formula

a(n) = A357274(n, 1).

cp's OEIS Frontend

A106506 Length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).

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A229849 Consider all primitive 120-degree triangles with sides A < B < C. The sequence gives the values of B.

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A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

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A321499 Numbers of the form (x - y)(x^2 - y^2) with x > y > 0.

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A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

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