cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A089025 Side of primitive equilateral triangle bearing at least one integral cevian that partitions an edge into two integral sections.

Original entry on oeis.org

8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112, 117, 119, 133, 143, 153, 160, 168, 171, 176, 187, 207, 209, 221, 224, 225, 247, 253, 255, 264, 275, 280, 285, 299, 312, 319, 323, 325, 341, 345, 352, 360, 377, 391, 403, 408, 416, 425, 435, 437, 440, 448
Offset: 1

Views

Author

Lekraj Beedassy, Nov 12 2003

Keywords

Comments

The cevians are numbers divisible only by primes of form 6n+1:A002476 (i.e., correspond to entries of A004611).
Composite cevians c belong to more than one equilateral triangle, actually to 2^(omega(c)-1) of them, where omega(n)=A001221(n). For instance, cevian 1813=7^2*37, with omega(1813)=2, belongs to 2^(2-1)=2 equilateral triangles, their sides being 1927=255+1627 and 1960=343+1617, while cevian 1729=7*13*19, with omega(1729)=3, belongs to 2^(3-1)=4 equilateral triangles whose sides are 1775=96+1679, 1824=209+1615, 1840=249+1591, 1859=299+1560.
Given a triangle with integer side lengths a, b, c relatively prime with a < b, c < b, and angle opposite c of 60 degrees then a*a - a*b + b*b = c*c from law of cosines and called a primitive Eisenstein triple by Gordon. This sequence is the possible side lengths of b. - Michael Somos, Apr 11 2012

Examples

			The equilateral triangle with side 280, for instance, has cevian 247 partitioning an edge into 93+187, as well as cevian 271 that sections the edge into 19+261.

Links

Crossrefs

Cf. A088513, A088514, A088977.

Programs

  • Mathematica
    findPrimIntEquiSide[maxC_] :=
Reap[Do[Do[
     With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
      If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
       Sow[c]; Break[]]], {a, Floor[c/2],
      1, -1}], {c, maxC}]][[2, 1]]
(* Andrew Turner, Aug 04 2017 *)

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

Views

Author

Bernard Schott, Sep 23 2022

Keywords

Comments

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).

Examples

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

Crossrefs

Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).
Cf. also A002324, A050931, A088514, A106505, A229849.

Programs

  • Maple
    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).

A088587 Difference between the two segments of edge d = A089025(n) partitioned by cevian c = A088513(n).

Original entry on oeis.org

2, 1, 11, 13, 26, 22, 23, 47, 13, 46, 11, 74, 61, 2, 37, 71, 107, 97, 23, 22, 118, 59, 146, 37, 143, 1, 109, 166, 191, 73, 83, 193, 26, 131, 94, 157, 11, 122, 239, 253, 59, 299, 73, 142, 286, 167, 169, 227, 362, 46, 121, 83, 277, 358, 218
Offset: 1

Views

Author

Lekraj Beedassy, Nov 20 2003

Keywords

Comments

a(n) is minimal, i.e., equals 1 for values c = A001570(n), d = A028230(n).

Crossrefs

Cf. A088514.

Programs

  • Mathematica
    findPrimIntEquiSectionDiff[maxC_] :=
Reap[Do[Do[
     With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
      If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
       Sow[c - 2 a]; Break[]]], {a, Floor[c/2],
      1, -1}], {c, maxC}]][[2, 1]]
(* Andrew Turner, Aug 04 2017 *)

Formula

a(n) = sqrt(4c^2 - 3d^2).

Extensions

a(35)=94 and a(36)=157 order reversed by Andrew Turner, Aug 04 2017
Showing 1-3 of 3 results.

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