A081961 -id:A081961 - cp's OEIS frontend

A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.

1, 3, 6, 10, 15, 16, 21, 22, 24, 28, 36, 37, 39, 42, 45, 46, 48, 51, 55, 58, 66, 67, 69, 72, 76, 78, 79, 84, 88, 91, 94, 97, 105, 106, 108, 111, 115, 120, 121, 123, 126, 130, 135, 136, 139, 142, 153, 154, 157, 163, 168, 171, 172, 174, 177, 181, 186, 190, 192, 193

Offset: 1

Hugo Pfoertner, Sep 13 2024

nonn

Sorted distinct values of ({A081961} - 1)/4.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A008846, A081961, A376208.

\\ Uses function is_a376208 from A376208
is_a376208(n,1)

from math import gcd, isqrt
test_all_k_upto   = 193
A375750, x, limit = set(), 2, test_all_k_upto * 4 + 1
while x**2 + (lowY := isqrt(2*x**2)-x)**2 < limit:
   for y in range(min(x-1,(yy:=isqrt(limit - x**2))-(yy%2 == x%2)), lowY,-2):
       if gcd(x,y) == 1: A375750.add(((x**2 + y**2) - 1) // 4)
   x += 1
print(A375750:=sorted(A375750)) # Karl-Heinz Hofmann, Sep 17 2024

A087938 Prime hypotenuse of primitive Pythagorean triangles with odd short leg (or even long leg).

Original entry on oeis.org

5, 13, 41, 61, 89, 97, 113, 149, 157, 181, 193, 233, 269, 277, 313, 317, 337, 353, 389, 421, 433, 461, 521, 541, 557, 569, 613, 617, 653, 673, 709, 761, 769, 773, 797, 821, 853, 881, 929, 937, 953, 1009, 1013, 1069, 1097, 1109, 1117, 1129, 1201, 1213, 1217

Offset: 1

Lekraj Beedassy, Oct 27 2003

nonn

Ordered intersection of A002144 and A081961.

These same triangles have the property that sqrt(h+b) = t, where h = a(n), b is the even long leg, t is an odd integer, and the odd short leg is an odd multiple of t, including some instances where that multiple is 1 (i.e., t equals the short leg). A similar property applies to the triangles of A087937. Also see A008846. - Richard R. Forberg, May 11 2016

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A002144, A008846, A081961, A087937.

Corrected and extended by Ray Chandler, Oct 28 2003

cp's OEIS Frontend

A375750 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an odd short leg.

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