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A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.

%I A344083 #10 May 09 2021 12:47:32
%S A344083 6,9,7,11,9,9,12,10,9,10,9,11,18,10,16,9,9,20,9,7,18,9,18,15,11,14,7,
%T A344083 12,10,13,12,7,12,15,12,17,14,18,13,9,13,14,15,10,9,7,9,21,12,10,15,
%U A344083 23,7,9,12,20,9,18,17,28,14,16,7,21,18,24,21,21,20,16,25
%N A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.
%C A344083 6 is the minimum possible value, and A176775(3,4,5) gives this minimum.
%C A344083 Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms.
%H A344083 Michel Marcus, <a href="/A344083/b344083.txt">Table of n, a(n) for n = 1..12471</a> (hypotenuses <= 10000).
%o A344083 (PARI) tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
%o A344083 itp(n) = my(m=tp(n)); (m-4+sqrtint((m-4)^2+8*(m-2)*n)) / (2*m-4); \\ A176775
%o A344083 f(v) = vecsum(apply(itp, v));
%o A344083 list(lim) = {my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [h, middle, small]);););); v = vecsort(Vec(v)); apply(f, v);}
%Y A344083 Cf. A176774, A176775, A342491.
%Y A344083 Cf. A090466, A090467.
%K A344083 nonn
%O A344083 1,1
%A A344083 _Michel Marcus_, May 09 2021