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 1-10 of 15 results. Next

A198453 Consider triples a<=b

Original entry on oeis.org

2, 2, 3, 3, 5, 6, 4, 9, 10, 5, 6, 8, 5, 14, 15, 6, 9, 11, 6, 20, 21, 7, 27, 28, 8, 10, 13, 8, 35, 36, 9, 13, 16, 9, 21, 23, 9, 44, 45, 10, 26, 28, 10, 54, 55, 11, 14, 18, 11, 20, 23, 11, 65, 66, 12, 17, 21, 12, 24, 27
Offset: 1

Views

Author

Charlie Marion, Oct 25 2011

Keywords

Comments

The definition can be generalized to define Pythagorean k-triples a<=b
If a, b and c form a Pythagorean k-triple, then n*a, n*b and n*c form a Pythagorean n*k-triple.
A triangle is defined to be a Pythagorean k-triangle if its sides form a Pythagorean k-triple.
If a, b and c are the sides of a Pythagorean k-triangle ABC with a<=b0 and acute when k<0. When k=0, the triangles are Pythagorean, as in the Beiler reference and Ron Knott's link.
For all k, the area of a Pythagorean k-triangle ABC with a<=b
Define a Pythagorean k-triple to be primitive if and only if there are no integers r>1, s>0 such that = for some Pythagorean s-triple . Thus, every Pythagorean 1-triple is primitive. For every k>1, the set of Pythagorean k-triples contains some non-primitive triples.
In particular, for d a proper divisor of k, it includes (k/d)*(a,b,c), where (a,b,c) is a Pythagorean d-triple. - Franklin T. Adams-Watters, Dec 01 2011

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

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

Links

Crossrefs

Cf. A103606, A198454-A198469.

A198456 Consider triples a<=b

Original entry on oeis.org

3, 6, 10, 8, 15, 11, 21, 28, 13, 36, 16, 23, 45, 28, 55, 18, 23, 66, 21, 27, 78, 46, 91, 20, 23, 36, 53, 105, 26, 41, 120, 136, 28, 52, 77, 153, 31, 58, 86, 171, 40, 49, 190, 33, 44, 54, 71, 210, 36, 41, 78, 116
Offset: 1

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.
The definition amounts to saying that T_a+T_b=T_c where T_i denotes a triangular number (A000217). - N. J. A. Sloane, Apr 01 2020

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

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

Links

Crossrefs

Cf. A000217, A156682, A198453-A198455, A198457-A198469, A333530, A333531.

A198455 Consider triples a<=b

Original entry on oeis.org

2, 5, 9, 6, 14, 9, 20, 27, 10, 35, 13, 21, 44, 26, 54, 14, 20, 65, 17, 24, 77, 44, 90, 14, 18, 33, 51, 104, 21, 38, 119, 135, 22, 49, 75, 152, 25, 55, 84, 170, 35, 45, 189, 26, 39, 50, 68, 209, 29, 35, 75, 114, 230, 125
Offset: 1

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.
The definition amounts to saying that T_a+T_b=T_c where T_i denotes a triangular number (A000217). - N. J. A. Sloane, Apr 01 2020

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

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

Links

Crossrefs

Cf. A000217, A156681, A198453, A198454, A198456-A198469, A333530, A333531.

A198454 Consider triples a<=b

Original entry on oeis.org

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

Views

Author

Charlie Marion, Oct 26 2011

Keywords

Comments

See A198453.

Examples

			2*3 + 2*3 = 3*4
3*4 + 5*6 = 6*7
4*5 + 9*10 = 10*11
5*6 + 6*7 = 8*9
5*6 + 14*15 = 15*16
6*7 + 9*10 = 11*12

References

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

Links

Crossrefs

Cf. A009004, A198453-A198469.

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.
Showing 1-10 of 15 results. Next

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

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