A381336 -id:A381336 - cp's OEIS frontend

A381337 a(n) is the smallest c >= A381336(n) + n for which a nondegenerate integer-sided triangle (A381336(n), A381336(n) + n, c) with an integer area exists.

5, 10, 15, 20, 25, 30, 13, 20, 15, 50, 25, 60, 41, 26, 75, 40, 25, 30, 29, 50, 35, 26, 37, 30, 39, 52, 45, 52, 109, 82, 41, 80, 55, 50, 65, 60, 61, 58, 61, 68, 73, 70, 65, 52, 75, 52, 53, 60, 61, 78, 75, 104, 203, 90, 75, 70, 87, 68, 101, 150, 89, 82, 91, 80, 117

Offset: 1

Felix Huber, Mar 18 2025

nonn

			a(5) = 25 because A381336(n) = 12 and the nondegenerate integer-sided triangle (12, 12 + 5, 25 >= 12 + 5) has an integer area (90), and there is no smaller c > 12 + 5 than 25 that satisfies this condition.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A381336.

A381337:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return c
            fi
        od
    od;
end proc;
seq(A381337(n),n=1..65);

A382297 Indices of right triangles in A381337.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 12, 14, 17, 23, 28, 31, 34, 35, 49, 51, 62, 69, 71, 73, 77, 85, 93, 97, 98, 102, 119, 127, 142, 161, 170, 194, 196, 199, 223, 233, 238, 241, 245, 279, 281, 287, 291, 337, 357, 381, 388, 391, 398, 439, 446, 449, 476, 482, 483, 511, 521, 527, 562

Offset: 1

Felix Huber, Mar 26 2025

nonn

A381336(a(n)) is the short leg, a(n) + A381336(a(n)) is the long leg and A381337(a(n)) is the hypotenuse.

			12 is in the sequence because A381336(12)^2 + (A381336(12) + 12)^2 = 36^2 + 48^2 = 60^2 = A381337(12)^2.

Cf. A381336, A381337.

isA382297:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return [k,n+k,c];
            fi
        od
    od;
end proc;
A382297:=proc(n)
    option remember;
    local a;
    if n=1 then
        1
    else
        for a from procname(n-1)+1 do
            if isA382297(a)[1]^2+isA382297(a)[2]^2=isA382297(a)[3]^2 then
                return a
            fi
        od
    fi;
end proc;
seq(A382297(n),n=1..59);

A381336(a(n))^2 + (A381336(a(n)) + a(n))^2 = A381337(a(n))^2.

cp's OEIS Frontend

A381337 a(n) is the smallest c >= A381336(n) + n for which a nondegenerate integer-sided triangle (A381336(n), A381336(n) + n, c) with an integer area exists.

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