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 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

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)

Links

Crossrefs

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

Extensions

Corrected by T. D. Noe, Jun 17 2004

A123323 Number of integer-sided triangles with maximum side n, with sides relatively prime.

Original entry on oeis.org

1, 1, 3, 4, 8, 7, 15, 14, 21, 20, 35, 26, 48, 39, 52, 52, 80, 57, 99, 76, 102, 95, 143, 100, 160, 132, 171, 150, 224, 148, 255, 200, 250, 224, 300, 222, 360, 279, 348, 296, 440, 294, 483, 370, 444, 407, 575, 392, 609, 460, 592, 516, 728, 495, 740, 588, 738, 644
Offset: 1

Views

Author

Franklin T. Adams-Watters, Sep 25 2006

Keywords

Comments

Number of triples a,b,c with a <= b <= c < a+b, gcd(a,b,c) = 1 and c = n.
Dropping the requirement for side lengths to be relatively prime this sequence becomes A002620 (with a different offset). See the Sep 2006 comment in A002620. - Peter Munn, Jul 26 2017

Links

Crossrefs

Cf. A002620, A051493, A054875, A070110, A123324, A123325, A239246.

Programs

  • Maple
    with(numtheory):
a:= n-> add(mobius(n/d)*floor((d+1)^2/4), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    a[n_] := DivisorSum[n, Floor[(#+1)^2/4]*MoebiusMu[n/#]&]; Array[a, 60] (* Jean-François Alcover, Dec 07 2015 *)
  • PARI
    A123323(n)=sumdiv(n,d,floor((d+1)^2/4)*moebius(n/d))

Formula

Moebius transform of b(n) = floor((n+1)^2/4).
G.f.: (G(x)+x-x^2)/2, where G(x) = Sum_{k >= 1} mobius(k)*x^k*(1+2*x^k-x^(2*k))/(1-x^k)^2/(1-x^(2*k)).

A054875 Number of pairwise incongruent triangles with integer sides and positive integer area and longest side of length n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 0, 3, 1, 3, 1, 0, 3, 2, 0, 0, 3, 4, 4, 0, 1, 2, 7, 0, 1, 0, 3, 3, 2, 5, 0, 6, 7, 4, 4, 0, 2, 4, 0, 0, 5, 0, 6, 7, 10, 4, 1, 3, 4, 0, 4, 0, 10, 3, 0, 3, 1, 11, 3, 0, 8, 1, 5, 0, 4, 6, 7, 13, 0, 3, 9, 0, 10, 0, 6, 0
Offset: 1

Views

Author

Henry Bottomley, May 26 2000

Keywords

Links

Crossrefs

Cf. A054876, A055592, A070787, A123323, A239246, A322105.

Programs

  • Mathematica
    okQ[x_, y_, z_] := If[x + y <= z, False, Module[{s = (x + y + z)/2}, IntegerQ[ Sqrt[s(s-x)(s-y)(s-z)]]] ]; a[n_] := Module[{num = 0}, Do[Do[If[okQ[x,y,n], num++], {x,1,y}], {y,1,n}]; num]; Array[a, 100] (* Amiram Eldar, Nov 22 2018 *)

Extensions

Name and offset changed to align with 2012 changes to A054876 by Peter Munn, Nov 23 2018

A096467 Numbers that can be the longest side of a primitive Heronian triangle.

Original entry on oeis.org

5, 6, 8, 13, 15, 17, 20, 21, 24, 25, 26, 28, 29, 30, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 73, 74, 75, 77, 80, 82, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113
Offset: 1

Views

Author

T. D. Noe, Jun 22 2004

Keywords

Comments

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.

Examples

			5 is on this list because the triangle with sides 3, 4, 5 has integral area.

Links

Crossrefs

Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).
Cf. A239246, A306626.

Programs

  • Mathematica
    nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

A306626 Numbers that set a record for occurrences as longest side of a primitive Heronian triangle.

Original entry on oeis.org

1, 5, 13, 17, 37, 52, 65, 85, 119, 125, 145, 221, 325, 481, 697, 725, 1025, 1105, 1625, 1885, 2465, 2665, 3145, 5525, 6409, 15457, 15725, 26129, 27625, 38425, 40885, 45305, 58565, 67405, 69745, 83317, 128945, 160225, 204425, 226525, 237133, 292825, 348725
Offset: 1

Views

Author

Amiram Eldar and Peter Munn, Mar 01 2019

Keywords

Comments

Congruent triangles are identified, that is to say mirror images are not distinguished.
The corresponding numbers of occurrences are 0, 1, 2, 3, 5, 6, 8, ...
A239246(k) gives the number of occurrences for any integer k.
The qualifier "primitive" means that we count only triangles whose sides have a gcd of 1. The equivalent sequence without this qualification is A322105.
The terms that are common with A322105 are 1, 5, 13, 52, 65, 145, 325, 1105, 5525, ...
The odd prime factors of the terms are almost all congruent to 1 modulo 4. a(9) = 119 = 7 * 17 provides the only exception in the first 50 terms. [updated by Peter Munn, Dec 04 2019]

Examples

			13 is in the sequence since it occurs in a record number of 2 triangles of side lengths {5, 12, 13} and {10, 13, 13}.
The side lengths, a(n), and their corresponding record numbers of occurrences, A239246(a(n)), are:
n       a(n)     prime factorization of a(n)  occurrences
1          1     -                               0
2          5     5                               1
3         13     13                              2
4         17     17                              3
5         37     37                              5
6         52     2^2 * 13                        6
7         65     5 * 13                          8
8         85     5 * 17                          9
9        119     7 * 17                         10
10       125     5^3                            13
11       145     5 * 29                         20
12       221     13 * 17                        30
13       325     5^2 * 13                       37
14       481     13 * 37                        42
15       697     17 * 41                        50
16       725     5^2 * 29                       54
17      1025     5^2 * 41                       63
18      1105     5 * 13 * 17                    90
19      1625     5^3 * 13                       93
20      1885     5 * 13 * 29                   106
21      2465     5 * 17 * 29                   116
22      2665     5 * 13 * 41                   134
23      3145     5 * 17 * 37                   178
24      5525     5^2 * 13 * 17                 277
25      6409     13 * 17 * 29                  373
26     15457     13 * 29 * 41                  396
27     15725     5^2 * 17 * 37                 463

Links

Crossrefs

Cf. A096467, A120130, A239246, A322105.

Programs

  • Mathematica
    okQ[x_, y_, z_] := GCD[x, y, z]==1 && If[x + y <= z, False, Module[{s = (x + y + z)/2}, IntegerQ[ Sqrt[s(s-x)(s-y)(s-z)]]] ]; a[n_] := Module[{num = 0}, Do[Do[If[okQ[x, y, n], num++], {x, 1, y}], {y, 1, n}]; num]; amax=-1; s={}; Do[a1=a[n]; If[a1 > amax, AppendTo[s, n]; amax=a1], {n, 1, 100}]; s

Extensions

a(28)-a(43) from Giovanni Resta, Nov 07 2019
Showing 1-5 of 5 results.

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