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 1-3 of 3 results.

A098996 a(n) = p*(p + 1)*(2*p + 1) where p is the n-th prime.

Original entry on oeis.org

30, 84, 330, 840, 3036, 4914, 10710, 14820, 25944, 51330, 62496, 105450, 142926, 164604, 214320, 306234, 421260, 465186, 615060, 731016, 794094, 1004880, 1164324, 1433790, 1853670, 2091306, 2217384, 2484540, 2625810, 2924214, 4145280, 4547796, 5199150, 5429340
Offset: 1

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Author

Parthasarathy Nambi, Nov 05 2004

Keywords

Comments

The unique primitive Pythagorean triple whose inradius is the n-th prime p and whose short leg is an odd number is (2*p+1, 2*p*(p+1), 2*p*(p+1)+1) and its area is a(n) = p*(p+1)*(2*p+1). - Miguel-Ángel Pérez García-Ortega, Mar 16 2025

Examples

			a(4) = 7*(7+1)*(2*7+1) = 840.

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Links

Crossrefs

Cf. A034953, A367573, A382097, A382070.

Programs

  • Magma
    [(p*(p+1)*(2*p+1)): p in PrimesUpTo(200)]; // Vincenzo Librandi, Feb 04 2011
  • Mathematica
    #(#+1)(2#+1)&/@Prime[Range[30]] (* Harvey P. Dale, Jun 23 2020 *)

Formula

a(n) = prime(n)*(prime(n)+1)*(2*prime(n)+1).

Extensions

More terms from Klaus Brockhaus, Nov 09 2004

A382070 Semiperimeter of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

Original entry on oeis.org

15, 28, 66, 120, 276, 378, 630, 780, 1128, 1770, 2016, 2850, 3486, 3828, 4560, 5778, 7140, 7626, 9180, 10296, 10878, 12720, 14028, 16110, 19110, 20706, 21528, 23220, 24090, 25878, 32640, 34716, 37950, 39060, 44850, 46056, 49770, 53628, 56280, 60378
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, Mar 15 2025

Keywords

Examples

			For n=2, the short leg is A367573(2,1) = 7, the long leg is A367573(2,2) = 24 and the hypotenuse is A367573(2,3) = 25 so the semiperimeter is then a(2) = (7 + 24 + 25)/2 = 28.

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Links

Crossrefs

Cf. A034953, A367573, A382097, A098996.

Programs

  • Mathematica
    a=Table[Prime[n],{n,1,40}];Apply[Join,Map[{(#+1)(2#+1)}&,a]]

Formula

a(n) = (prime(n)+1) * (2*prime(n)+1).
a(n) = (A367573(n,1) + A367573(n,2) + A367573(n,3))/2.

A382097 Sum of the legs of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

Original entry on oeis.org

17, 31, 71, 127, 287, 391, 647, 799, 1151, 1799, 2047, 2887, 3527, 3871, 4607, 5831, 7199, 7687, 9247, 10367, 10951, 12799, 14111, 16199, 19207, 20807, 21631, 23327, 24199, 25991, 32767, 34847, 38087, 39199, 44999, 46207, 49927, 53791, 56447, 60551
Offset: 1

Views

Author

Miguel-Ángel Pérez García-Ortega, Mar 15 2025

Keywords

Examples

			For n=2, the short leg is A367573(2,1) = 7 and the long leg is A367573(2,2) = 24 so the sum of the legs is then a(2) = 7 + 24 = 31.

References

  • Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Links

Crossrefs

Cf. A034953, A367573, A382070, A098996.

Programs

  • Mathematica
    a=Table[Prime[n],{n,1,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

a(n) = 2*(prime(n))^2 + 4*prime(n) + 1.
a(n) = A367573(n,1) + A367573(n,2).
Showing 1-3 of 3 results.

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