A379596 a(n) is the least positive integer k for which k^2 + (k + n)^2 is a square.

3, 6, 9, 12, 15, 18, 5, 24, 27, 30, 33, 36, 39, 10, 45, 48, 7, 54, 57, 60, 15, 66, 12, 72, 75, 78, 81, 20, 87, 90, 9, 96, 99, 14, 25, 108, 111, 114, 117, 120, 36, 30, 129, 132, 135, 24, 16, 144, 11, 150, 21, 156, 159, 162, 165, 40, 171, 174, 177, 180, 183, 18, 45

Offset: 1

Author

Felix Huber, Feb 15 2025

Keywords

• nonn
• easy

Comments

a(n) is also the smallest short leg of a Pythagorean triangle where the difference between the two legs is n.

A289398(n) is the least integer m > n for which (n^2 + m^2)/2 is a square. This is equivalent to the least positive integer k for which (n^2 + (n + 2*k)^2)/2 = k^2 + (n + k)^2 is a square. From m = n + 2*k follows a(n) = (A289398(n) - n)/2.

Examples

			a(1) = 3 because 3^2 + (3 + 1)^2 = 5^2 and there is no smaller positive integer k than 3 with that property.
a(28) = 20 because 20^2 + (20 + 28)^2 = 52^2 and there is no smaller positive integer k than 20 with that property.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A089015, A009004, A120682, A132404, A289398, A379830.

Programs

Maple

A379596:=proc(n)
    local k;
    for k do
        if issqr(k^2+(k+n)^2) then
            return k
        fi
    od
end proc;
seq(A379596(n),n=1..63);

Mathematica

s={};Do[k=0;Until[IntegerQ[Sqrt[k^2+(k+n)^2]],k++];AppendTo[s,k],{n,63}];s (* James C. McMahon, Mar 02 2025 *)

PARI

a(n) = my(k=1); while (!issquare(k^2 + (k + n)^2), k++); k; \\ Michel Marcus, Feb 15 2025

Python

from itertools import count
from sympy.ntheory.primetest import is_square
def A379596(n): return next(k for k in count(1) if is_square(k**2+(k+n)**2)) # Chai Wah Wu, Mar 02 2025

Formula

a(n) = (A289398(n) - n)/2.