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

Author

Miguel-Ángel Pérez García-Ortega, Mar 01 2024

Keywords

• nonn
• easy
• tabf

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

Examples

			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;
  ...
		

References

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

Links

• Miguel-Ángel Pérez García-Ortega, Capítulo 5.

Crossrefs

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

Programs

Mathematica

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

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).

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