A370763 Table read by rows: row n is the unique primitive Pythagorean triple (a,b,c) such that a = prime(n)*prime(n+1) and its long leg and hypotenuse are consecutive natural numbers.

15, 112, 113, 35, 612, 613, 77, 2964, 2965, 143, 10224, 10225, 221, 24420, 24421, 323, 52164, 52165, 437, 95484, 95485, 667, 222444, 222445, 899, 404100, 404101, 1147, 657804, 657805, 1517, 1150644, 1150645, 1763, 1554084, 1554085, 2021, 2042220, 2042221, 2491, 3102540, 3102541

Offset

2,1

Comments

The pair of natural numbers (d,e) is said to be a pair of primitive twin divisors of a natural number m when d*e = m and gcd(d,e) = 1.

Given two prime numbers p and q (p<q), as the only pairs of primitive twin divisors of (p*q)^2 are (1,p^2*q^2) and (p^2,q^2), then, the only primitive Pythagorean triples with a short leg equal to p*q are (p*q,(p^2*q^2-1)/2,(p^2*q^2+1)/2) and (p*q,(q^2-p^2)/2,(q^2+p^2)/2), being (p*q,(p^2*q^2-1)/2,(p^2*q^2+1)/2) the only one that has consecutive long leg and hypotenuse.

References

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Table of n, a(n) for n=2..43.

Miguel-Ángel Pérez García-Ortega, Capítulo 5.

Formula

Row n = (a, b, c) = (p * q, (p^2 * q^2 - 1)/2, (p^2 * q^2 + 1)/2), where p = prime(n) and q = prime(n+1).

Example

Table begins:
  n=2:   15,   112,   113;
  n=3:   35,   612,   613;
  n=4:   77,  2964,  2965;
  n=5:  143, 10224, 10225;
  n=6:  221, 24420, 24421;
  ...

Mathematica

Apply[Join, Map[{#, (#^2-1)/2, (#^2+1)/2} &, Prime[Range[2, 31]]Prime[Range[3, 32]]]]

Crossrefs

Cf. A000040, A006094 (short leg), A102770 (inradius).

Sequence in context: A105051 A105040 A298123 * A295384 A110822 A222410

Adjacent sequences: A370760 A370761 A370762 * A370764 A370765 A370766

Keyword

nonn,easy,tabf

Author

Miguel-Ángel Pérez García-Ortega, Mar 01 2024

Status

approved