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 1-7 of 7 results.

A086909 Middle side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

4, 26, 28, 52, 76, 98, 124, 134, 158, 148, 172, 206, 218, 266, 244, 316, 292, 362, 388, 388, 364, 364, 386, 398, 518, 556, 494, 532, 556, 508, 532, 602, 602, 628, 724, 676, 758, 746, 734, 854, 916, 806, 868, 916, 844, 892, 866, 868, 1036, 1022, 988, 964, 974
Offset: 1

Views

Author

Lekraj Beedassy, Sep 19 2003

Keywords

References

  • J. A. MacDougall, "Heron Triangles With Sides In Arithmetic Progression", Journal of Recreational Mathematics 31(3) 2002-2003, pp. 192-194.

Links

Crossrefs

Cf. A072330, A072360, A089019, A089020, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n==1, 4, For[b = 2d, True, b++, a = b-d; c = b+d; p = (a+b+c)/2; If[IntegerQ[p] && IntegerQ[ Sqrt[p(p-a)(p-b)(p-c)]] && GCD[a, b, c] == 1, Return[b]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 06 2020 *)

Extensions

Extended by Ray Chandler, Jul 03 2004

A072330 Common difference n such that primitive triangles exist which are n-arithmetic (i.e., primitive Heronian triangles whose sides in arithmetic progression have common difference n).

Original entry on oeis.org

1, 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 121, 131, 143, 157, 167, 169, 179, 181, 191, 193, 227, 229, 239, 241, 251, 253, 263, 277, 299, 311, 313, 337, 347, 349, 359, 373, 383, 397, 407, 409, 419, 421, 431, 433, 443, 457, 467, 479, 481, 491, 503
Offset: 1

Views

Author

Lekraj Beedassy, Jul 15 2002

Keywords

Comments

The first entry in particular is associated with sequences A003500 and A007655.
Such a triangle has a middle side 2*x partitioned into x +- 2*n by the corresponding altitude (i.e., median and altitude points are always a distance 2*n apart).

Links

Crossrefs

Cf. A072360, A086909, A089019, A089020, A096672, A096673, A096674.

Programs

  • Maple
    isA072330 := proc(n)
    if n = 1 then
        true;
    else
        for p in ifactors(n)[2] do
            if not modp(op(1,p),12) in {1,11} then
                return false ;
            end if;
        end do:
        true;
    end if;
end proc:
for n from 1 to 1000 do
    if isA072330(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 26 2017
  • Mathematica
    fac12Q[n_] := And @@ (MemberQ[{1, 11}, #] & /@ Mod[First /@ FactorInteger@ n, 12]); Select[Range[600], fac12Q] (* Frank M Jackson, Apr 09 2016 with simplification by Giovanni Resta *)
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
Select[Range[1000], okQ] (* Jean-François Alcover, Mar 06 2020 *)

Formula

n = 1 or a product of primes p congruent to +- 1 (mod 12).

Extensions

Corrected and extended by Ray Chandler, Jul 02 2004
Incorrect b-file by Carmine Suriano replaced by Frank M Jackson, May 09 2016

A072360 One-sixth the area of the smallest primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

1, 26, 21, 91, 95, 196, 341, 536, 790, 259, 559, 1030, 654, 2926, 549, 4029, 1241, 4706, 5529, 5335, 1729, 1001, 1544, 2786, 9324, 12649, 4446, 8645, 9591, 1651, 3059, 10234, 3010, 3925, 19005, 2535, 16676, 14174, 8074, 25620, 33205, 8060
Offset: 1

Views

Author

Lekraj Beedassy, Jul 18 2002

Keywords

Comments

Such a triangle has middle side 2*x'.

Links

Crossrefs

Cf. A072330, A086909, A089019, A089020, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p, area}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n == 1, 1, For[b = 2 d, True, b++, a = b - d; c = b + d; p = (a + b + c)/2; If[IntegerQ[p] && IntegerQ[area = Sqrt[p (p - a) (p - b) (p - c)]] && GCD[a, b, c] == 1, Return[area/6]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 07 2020 *)

Formula

a(n) = x'*y'/2, where (x', y') is the fundamental solution to x^2 - 3*y^2 = d^2, where d=A072330(n).

Extensions

Edited, corrected and extended by Ray Chandler, Jul 03 2004

A089020 First value y satisfying x^2 - 3*y^2 = d^2, where d=A072330(n).

Original entry on oeis.org

1, 4, 3, 7, 5, 8, 11, 16, 20, 7, 13, 20, 12, 44, 9, 51, 17, 52, 57, 55, 19, 11, 16, 28, 72, 91, 36, 65, 69, 13, 23, 68, 20, 25, 105, 15, 88, 76, 44, 120, 145, 40, 87, 119, 29, 85, 24, 17, 155, 132, 93, 31, 44, 104, 52, 92, 95, 19, 140, 200, 28, 105, 231, 35, 100, 185, 105, 120
Offset: 1

Views

Author

Lekraj Beedassy, Nov 04 2003

Keywords

Links

Crossrefs

For corresponding x see A089019.
Cf. A072330, A072360, A086909, A089019, A096672, A096673, A096674.

Programs

  • Mathematica
    terms = 1000;
nmax = 12 terms;
okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], MatchQ[Mod[#, 12], 1|11]&];
A072330 = Select[Range[nmax], okQ];
a[n_] := Module[{a, b, c, d, p}, d = If[n <= Length[A072330], A072330[[n]], Print["nmax = ", nmax, " insufficient"]; Exit[]]; If[n == 1, 1, For[b = 2 d, True, b++, a = b - d; c = b + d; p = (a + b + c)/2; If[IntegerQ[p] && IntegerQ[Sqrt[p (p - a) (p - b) (p - c)]] && GCD[a, b, c] == 1, Return[ Sqrt[b^2 - 4 d^2]/(2 Sqrt[3])]]]]];
a /@ Range[terms] (* Jean-François Alcover, Mar 07 2020 *)

Extensions

Extended by Ray Chandler, Jul 03 2004

A096672 Smallest side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

3, 15, 15, 29, 39, 51, 65, 73, 87, 75, 89, 109, 111, 157, 123, 185, 149, 205, 221, 219, 185, 183, 195, 205, 291, 327, 255, 291, 305, 255, 269, 325, 303, 317, 411, 339, 411, 397, 375, 481, 533, 409, 461, 507, 425, 471, 435, 435, 593, 565, 521, 485, 493, 555
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096673, A096674.

A096673 Largest side of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

5, 37, 41, 75, 113, 145, 183, 195, 229, 221, 255, 303, 325, 375, 365, 447, 435, 519, 555, 557, 543, 545, 577, 591, 745, 785, 733, 773, 807, 761, 795, 879, 901, 939, 1037, 1013, 1105, 1095, 1093, 1227, 1299, 1203, 1275, 1325, 1263, 1313, 1297, 1301, 1479, 1479
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096672, A096674.

A096674 Semiperimeter of the first primitive d-arithmetic triangle, where d=A072330(n).

Original entry on oeis.org

6, 39, 42, 78, 114, 147, 186, 201, 237, 222, 258, 309, 327, 399, 366, 474, 438, 543, 582, 582, 546, 546, 579, 597, 777, 834, 741, 798, 834, 762, 798, 903, 903, 942, 1086, 1014, 1137, 1119, 1101, 1281, 1374, 1209, 1302, 1374, 1266, 1338, 1299, 1302, 1554, 1533
Offset: 1

Views

Author

Ray Chandler, Jul 03 2004

Keywords

Crossrefs

Cf. A072330, A072360, A086909, A089019, A089020, A096672, A096673.
Showing 1-7 of 7 results.

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