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%I #47 Feb 17 2024 15:04:40
%S 13530,136,35,5,4510,10,100,45,51,1404
%N a(n) is the least integer h such that there exists a Pythagorean triple (x, y, h) that satisfies f(x)+f(y)+f(h)=n where f(m)=A176774(m) is the smallest polygonality of m; a(n) = 0 if no such h exists.
%C a(19) > 10^9 if it exists.
%C It appears that the triples whose sum is 10 (as in the 2nd example below) have legs n^6 = A001014(n), (n^8 - n^4)/2 = A218131(n+1)/2 and (n^8 + n^4)/2 = A071231(n) for n >= 2; they consist of 2 triangular numbers and 1 square number. - _Michel Marcus_, Apr 12 2021
%H Michel Marcus, <a href="/A342858/a342858.txt">Table of n, a(n) for n = 9..10000, with 0 for a(19)</a>
%e a(9) = 13530 with A176774([8778, 10296, 13530]) = [3,3,3].
%e a(10) = 136 with A176774([64, 120, 136]) = [4,3,3].
%e a(11) = 35 with A176774([21, 28, 35]) = [3,3,5].
%e a(12) = 5 with A176774([3, 4, 5]) = [3,4,5].
%e a(13) = 4510 with A176774([2926, 3432, 4510]) = [3,5,5].
%e a(14) = 10 with A176774([6, 8, 10]) = [3,8,3].
%e a(15) = 100 with A176774([28, 96, 100]) = [3,8,4].
%e a(16) = 45 with A176774([27, 36, 45]) = [10,3,3].
%e a(17) = 51 with A176774([45, 24, 51]) = [3,9,5].
%e a(18) = 1404 with A176774([540, 1296, 1404]) = [7,4,7].
%o (PARI) tp(n) = if (n<3, [n], my(v=List()); fordiv(2*n, k, if(k<2, next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v, s))); v = Vec(v); v[#v]); \\ A176774
%o vsum(v) = vecsum(apply(tp, v));
%o lista(limp, lim) = {my(vr = vector(limp)); for(u = 2, sqrtint(lim), for(v = 1, u, if (u*u+v*v > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u*u+v*v) > lim, break); my(w = [i*(u*u - v*v), i*2*u*v, i*(u*u+v*v)]); my(h = i*(u*u+v*v)); my(sw = vsum(w)); if (sw <= limp, if (vr[sw] == 0, vr[sw] = h, if (h < vr[sw], vr[sw] = h))););););); vector(#vr - 8, k, vr[k+8]);}
%o lista(80, 15000) \\ _Michel Marcus_, Apr 16 2021
%Y Cf. A009000, A046083, A046084, A176774.
%Y Cf. A213188 (see 2nd comment).
%Y Cf. A245646, A245647, A245648, A342491.
%Y Cf. A001014, A071231, A218131.
%K nonn,hard,more
%O 9,1
%A _Michel Marcus_, Mar 26 2021
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