A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

35, 41, 140, 164, 205, 221, 315, 369, 389, 391, 560, 656, 689, 775, 820, 875, 884, 1025, 1189, 1260, 1476, 1556, 1564, 1565, 1625, 1715, 1739, 1781, 1845, 1855, 1989, 2009, 2240, 2624, 2756, 2835, 3100, 3280, 3321, 3500, 3501, 3519, 3536, 3865, 3869, 4100, 4105

Offset: 1

Author

Felix Huber, Jan 18 2025

Keywords

• nonn

Comments

Corresponding long legs in A380073, short legs in A380074.

Subsequence of A009000 and supersequence of A088319.

Examples

			35 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.
206125 is twice in the sequence because 31525^2 + 203700^2 = 206125^2 and 31525 + 203700 = 485^2 as well as 94588^2 + 183141^2 = 206125^2 and 94588 + 183141 = 527^2.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10001
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A000290, A007913, A009000, A088319, A379830, A380073, A380074.

Programs

Maple

# Calculates the first 10001 terms
A380072:=proc(M)
    local i,m,p,q,r,w,L;
    L:=[];
    m:=M^2+2*M+2;
    for p from 2 to M do
        for q to p-1 do
            if gcd(p,q)=1 and (is(p,even) or is(q,even)) then
                r:=1;
                for i in ifactors(p^2-q^2+2*p*q)[2] do
                    if is(i[2],odd) then
                        r:=r*i[1]
                    fi
                od;
                w:=r*(p^2+q^2);
                if w<=m then
                    L:=[op(L),seq(i^2*w,i=1..floor(sqrt(m/w)))]
                fi
            fi
        od
    od;
    return op(sort(L))
end proc;
A380072(4330);