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 1-9 of 9 results.

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

A089019 First nontrivial x satisfying x^2 - 3*y^2 = d^2, where d=A072330(n).

Original entry on oeis.org

2, 13, 14, 26, 38, 49, 62, 67, 79, 74, 86, 103, 109, 133, 122, 158, 146, 181, 194, 194, 182, 182, 193, 199, 259, 278, 247, 266, 278, 254, 266, 301, 301, 314, 362, 338, 379, 373, 367, 427, 458, 403, 434, 458, 422, 446, 433, 434, 518, 511, 494, 482, 487, 523
Offset: 1

Views

Author

Lekraj Beedassy, Nov 04 2003

Keywords

Links

Crossrefs

For corresponding y see A089020.
Cf. A072330, A072360, A086909, A089020, A096672, A096673, A096674.

Programs

Formula

a(n) = A086909(n)/2.

Extensions

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.

A095267 Least perimeter common to 2^n primitive arithmetic triangles, i.e., Heronian triangles whose sides are in arithmetic progression.

Original entry on oeis.org

12, 84, 1092, 20748, 643188, 23797956, 1023312108, 62422038588, 4182276585396, 305306190733908, 24119189067978732, 2339561339593937004, 240974817978175511412, 26266255159621130743908
Offset: 0

Views

Author

Lekraj Beedassy, Jul 02 2004

Keywords

Examples

			1092 is the perimeter common to the 2^2=4 arithmetic triangles (183,364,545), (185,364,543), (257,364,471), (303,364,425) having common differences 181, 179, 107, 61 and areas 6006, 10376, 46410, 54054, respectively. (Middle side is the same in all triangles.)

Links

Crossrefs

Cf. A086909.

Formula

a(n) = 3*A095288(n) = a(n-1)*A002476(n). - Ray Chandler, Jul 09 2004

Extensions

More terms from Ray Chandler, Jul 09 2004

A095288 Least middle side of 2^n primitive arithmetic triangles, i.e., primitive Heronian triangles whose sides are in arithmetic progression.

Original entry on oeis.org

4, 28, 364, 6916, 214396, 7932652, 341104036, 20807346196, 1394092195132, 101768730244636, 8039729689326244, 779853779864645668, 80324939326058503804, 8755418386540376914636, 1111938135090627868158772
Offset: 0

Views

Author

Lekraj Beedassy, Jul 02 2004

Keywords

Examples

			6916 is the first middle side belonging to 2^3=8 arithmetic triangles, viz., (3485,6916,10347), (3747,6916,10085), (4133,6916,9699), (4299,6916,9533), (4827,6916,9005), (5045,6916,8787), (5333,6916,8499), (6267,6916,7565) of common differences 3431, 3169, 2783, 2617, 2089, 1871, 1583, 649 respectively and integral areas. (Perimeter is obviously three times middle side and hence the same for all the triangles.)

Crossrefs

Cf. A086909.

Formula

a(n) = A095267(n)/3 = a(n-1) * A002476(n). - Ray Chandler, Jul 09 2004

Extensions

More terms from Ray Chandler, Jul 09 2004
Showing 1-9 of 9 results.

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