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A343981 a(n) is the least integer h such that there exists a Pythagorean triple whose hypotenuse is h and whose other legs z satisfy A176774(z) = n.

%I A343981 #10 May 07 2021 02:15:30
%S A343981 35,0,13,0,2727,104,13911,17370,426996,1855,340119,89375,3588,37400,
%T A343981 3034,57709,2103750,88400,53290,506817,15263560,141921,660350,3372270,
%U A343981 419356,40716,57526469,356025,639135,5316785,872934,1493219,11939849,119616,331290,3008185
%N A343981 a(n) is the least integer h such that there exists a Pythagorean triple whose hypotenuse is h and whose other legs z satisfy A176774(z) = n.
%C A343981 a(4)=0 is conjectured.
%C A343981 a(6)=0 because all hexagonal numbers are triangular numbers (see A176948).
%H A343981 Michel Marcus, <a href="/A343981/b343981.txt">Table of n, a(n) for n = 3..70</a>
%H A343981 Michel Marcus, <a href="/A343981/a343981.txt">Triples for n = 3..70</a>
%e A343981 a(3)=35 because of [21, 28, 35] where A176774(21) = A176774(35) = 3.
%e A343981 a(5)=13 because of [5, 12, 13] where A176774(5) = A176774(12) = 5.
%e A343981 a(7)=2727 because of [540, 2673, 2727] where A176774(540) = A176774(2673) = 7.
%o A343981 (PARI) p(s, n) = ((s-2)*n^2 - (s-4)*n)/2;
%o A343981 lista(nn, n) = {my(v = vector(nn, k, p(n, k))); v = select(x->(tp(x)==n), v); my(kh = oo, kv = oo); for (i=1, #v, for (j=1, i, my(h2 = v[i]^2 + v[j]^2, h); if (issquare(h2, &h), if (h < kh, kh = h; kv = [v[j], v[i], kh]);););); kh;}
%o A343981 a(n) = {if (n==4, return (0)); if (n==6, return (0)); my(nn = 2); while ((res=lista(nn, n)) == oo, nn *= 2); res;}
%Y A343981 Cf. A176774, A176948.
%K A343981 nonn
%O A343981 3,1
%A A343981 _Michel Marcus_, May 06 2021