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.

Previous Showing 11-20 of 29 results. Next

A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

Original entry on oeis.org

18, 48, 60, 90, 100, 140, 144, 180, 210, 270, 280, 288, 294, 320, 360, 378, 448, 462, 480, 594, 600, 648, 660, 720, 728, 756, 858, 900, 900, 924, 980, 1008, 1008, 1078, 1080, 1120, 1170, 1210, 1260, 1344, 1496, 1530, 1530, 1568, 1584, 1584, 1680, 1700, 1728
Offset: 1

Views

Author

Alexandre Wajnberg, May 02 2005

Keywords

Examples

			a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.

Links

Crossrefs

Cf. A103605, A103606, A024364, A024406, A105521.

Programs

  • Mathematica
    L = {}; mx = 1728; Do[ Do[ If[ IntegerQ[z = Sqrt[x^2 + y^2]], v = x y/2 + x + y + z; If[v <= mx, AppendTo[L, v], Break[]]], {y, x-1}], {x, 4, 4 + (2 mx^2)^(1/3)}]; Sort@ L (* Giovanni Resta, Mar 16 2020 *)
  • Rexx
    T. = 0                        ;  S = ''
do C = 1 to 999               ;  H = C*C
   do D = 1 to C              ;  I = D*D
      do E = 1 to D           ;  J = E*E
         if I + J < H   then  iterate E
         if I + J = H   then  do
            K = T.0 + 1       ;  T.0 = K
            P = C + D + E     ;  A = ( D * E ) / 2
            T.K = right( A + P, 6 )
            T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
         end
         leave E
      end E
   end D
end C
call KWIK 'T.' /* sort by A+P for area A and perimeter P */
Y = 0
do N = 1 to T.0 while length( S ) < 255
   X = word( T.N, 1 )         ;  say T.N
   if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
   S = S || ', ' || X         ;  Y = X
end N
say substr( S, 3 )            /* Frank Ellermann, Mar 02 2020 */

Extensions

Corrected and extended by Frank Ellermann, Mar 02 2020

A198458 Consider triples a<=b

Original entry on oeis.org

3, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 28, 28, 29, 30, 30, 30, 30, 30, 30, 31, 31, 31
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198457, A198459-A198469.

A198459 Consider triples a<=b

Original entry on oeis.org

6, 4, 16, 10, 8, 30, 18, 14, 48, 12, 28, 70, 18, 40, 16, 30, 96, 25, 54, 22, 40, 126, 20, 33, 70, 160, 26, 42, 88, 24, 64, 198, 52, 108, 30, 78, 240, 28, 40, 63, 130, 54, 286, 34, 48, 75, 154, 32, 64, 110, 336, 88, 180, 38, 128, 390, 28, 36, 66, 102, 208, 448, 33, 42
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198458, A198460-A198469.

A198460 Consider triples a<=b

Original entry on oeis.org

7, 6, 17, 12, 11, 31, 20, 17, 49, 16, 30, 71, 22, 42, 21, 33, 97, 29, 56, 27, 43, 127, 26, 37, 72, 161, 32, 46, 90, 31, 67, 199, 56, 110, 37, 81, 241, 36, 46, 67, 132, 59, 287, 42, 54, 79, 156, 41, 69, 113, 337, 92, 182, 47, 131, 391, 40, 46, 72, 106, 210, 449, 45, 52
Offset: 1

Views

Author

Charlie Marion, Nov 15 2011

Keywords

Comments

See A198453 and A198457.

Examples

			3*5 + 6*8 = 7*9
4*6 + 4*6 = 6*8
5*7 + 16*17 = 17*18
6*8 + 10*12 12*14
7*9 + 8*10 = 11*13
7*9 + 30*32 = 31*33

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198459, A198461-A198469.

A198461 Consider triples a<=b

Original entry on oeis.org

2, 3, 4, 3, 7, 8, 4, 12, 13, 5, 18, 19, 6, 6, 9, 6, 11, 13, 6, 25, 26, 7, 15, 17, 7, 33, 34, 8, 42, 43, 9, 10, 14, 9, 15, 18, 9, 52, 53, 10, 30, 32, 10, 63, 64, 11, 36, 38, 11, 75, 76, 12, 14, 19, 12, 19, 23, 12, 27, 30, 12, 88, 89, 13, 102, 103, 14, 57, 59, 14, 117, 118
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198460, A198462-A198469.

A198462 Consider triples a<=b

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 6, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 15, 15, 16, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 27, 28
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198461, A198463-A198469.

A198463 Consider triples a<=b

Original entry on oeis.org

3, 7, 12, 18, 6, 11, 25, 15, 33, 42, 10, 15, 52, 30, 63, 36, 75, 14, 19, 27, 88, 102, 75, 117, 18, 23, 42, 65, 133, 150, 30, 39, 168, 22, 27, 60, 92, 187, 102, 207, 42, 54, 228, 22, 26, 31, 81, 250, 51, 135, 273, 147, 297, 30, 35, 105, 322, 45, 66, 84, 348
Offset: 1

Views

Author

Charlie Marion, Nov 26 2011

Keywords

Comments

See A198453.

Examples

			2*5 + 3*6 = 4*7
3*6 + 7*10 = 8*11
4*7 + 12*15 = 13*16
5*8 + 18*21 = 19*22
6*9 + 6*9 = 9*12
6*9 + 11*14 = 13*16

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, pp. 104-134.

Links

Crossrefs

Cf. A103606, A198453-A198462, A198464-A198469.

A334382 Least k whose set of divisors contains exactly n Pythagorean triples, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 360, 960, 720, 3840, 1440, 2160, 2880, 8160, 3600, 69360, 8400, 8640, 7200, 32640, 9360, 16800, 14400, 34560, 24480, 130560, 18720, 77760, 54600, 28080, 25200, 67200, 37440, 11045580, 61200, 73440, 97920, 294000, 46800, 65520, 50400, 268800, 109200
Offset: 1

Views

Author

Michel Lagneau, Apr 26 2020

Keywords

Comments

This is a subsequence of A169823: a(n) == 0 (mod 60) because one side of every Pythagorean triple is divisible by 3, another by 4, and another by 5. The smallest and best-known Pythagorean triple is (a, b, c) = (3, 4, 5).

Examples

			a(3) = 240 because the set of divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.

Links

Crossrefs

Cf. A103605, A103606, A169823, A334080.

Programs

  • Maple
    with(numtheory):
for n from 1 to 52 do :
ii:=0:
for k from 60 by 60 to 10^8 while(ii=0) do:
   d:=divisors(k):n0:=nops(d):it:=0:
    for i from 1 to n0-1 do:
     for j from i+1 to n0-2 do :
      for m from i+2 to n0 do:
       if d[i]^2 + d[j]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    if it = n
     then
     ii:=1: printf (`%d %d \n`,n,k):
     else
    fi:
od:
od:

Extensions

a(31) from Giovanni Resta, Apr 27 2020

A359073 Sum of square end-to-end displacements over all n-step self-avoiding walks of A359709.

Original entry on oeis.org

0, 4, 16, 44, 160, 556, 1744, 12252, 15840, 98876, 138160, 709900, 1155616, 5098260, 11820656, 37085908, 111147104, 281078764, 932893104, 2255139900, 7295211968, 18928121236, 54864568720, 160016686500, 404167501888, 1331607134172, 2945597090384, 10805511468852, 21448743511648
Offset: 0

Views

Author

Scott R. Shannon, Jan 12 2023

Keywords

Crossrefs

Cf. A359709, A103606, A359133, A336448, A001411, A356617, A173380, A337353, A358036, A358046.

A359709 Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.

Original entry on oeis.org

1, 4, 4, 12, 28, 76, 164, 732, 1044, 4924, 6724, 30636, 43972, 190516, 313996, 1197908, 2284260, 7678188, 16257604, 50524252, 113052396, 341811828, 773714436, 2358452388, 5245994292, 16447462492, 35395532236, 115129727188, 238542983748, 804980005276
Offset: 0

Views

Author

Scott R. Shannon, Jan 12 2023

Keywords

Comments

The walks counted are all those directly along and x or y axes, and all walks whose final (|x|,|y|) lattice point are the two legs of a Pythagorean triple.

Examples

			a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
     X---.
         |
     X---.
.
This can be walked in 8 different ways on a 2D square lattice. There are also the four walks directly along the x and y axes, giving a total of 8 + 4 = 12 walks.

Links

Crossrefs

Cf. A359073, A103606, A359741, A001411, A356617, A173380, A337353, A358036, A358046.
Previous Showing 11-20 of 29 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.