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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.
1
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
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
KEYWORD
nonn,easy,tabf
STATUS
approved