A342491 -id:A342491 - cp's OEIS frontend

A342858 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.

13530, 136, 35, 5, 4510, 10, 100, 45, 51, 1404

Offset: 9

Michel Marcus, Mar 26 2021

a(19) > 10^9 if it exists.

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

			a(9)  = 13530 with A176774([8778, 10296, 13530]) = [3,3,3].
a(10) = 136   with A176774([64, 120, 136])       = [4,3,3].
a(11) = 35    with A176774([21, 28, 35])         = [3,3,5].
a(12) = 5     with A176774([3, 4, 5])            = [3,4,5].
a(13) = 4510  with A176774([2926, 3432, 4510])   = [3,5,5].
a(14) = 10    with A176774([6, 8, 10])           = [3,8,3].
a(15) = 100   with A176774([28, 96, 100])        = [3,8,4].
a(16) = 45    with A176774([27, 36, 45])         = [10,3,3].
a(17) = 51    with A176774([45, 24, 51])         = [3,9,5].
a(18) = 1404  with A176774([540, 1296, 1404])    = [7,4,7].

Michel Marcus, Table of n, a(n) for n = 9..10000, with 0 for a(19)

Cf. A009000, A046083, A046084, A176774.
Cf. A213188 (see 2nd comment).
Cf. A245646, A245647, A245648, A342491.
Cf. A001014, A071231, A218131.

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
vsum(v) = vecsum(apply(tp, v));
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]);}
lista(80, 15000) \\ Michel Marcus, Apr 16 2021

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.

Original entry on oeis.org

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, 12, 10, 13, 12, 7, 12, 15, 12, 17, 14, 18, 13, 9, 13, 14, 15, 10, 9, 7, 9, 21, 12, 10, 15, 23, 7, 9, 12, 20, 9, 18, 17, 28, 14, 16, 7, 21, 18, 24, 21, 21, 20, 16, 25

Offset: 1

Michel Marcus, May 09 2021

nonn

6 is the minimum possible value, and A176775(3,4,5) gives this minimum.

Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms.

Michel Marcus, Table of n, a(n) for n = 1..12471 (hypotenuses <= 10000).

Cf. A176774, A176775, A342491.

Cf. A090466, A090467.

tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
itp(n) = my(m=tp(n)); (m-4+sqrtint((m-4)^2+8*(m-2)*n)) / (2*m-4); \\ A176775
f(v) = vecsum(apply(itp, v));
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);}

cp's OEIS Frontend

A342858 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.

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