cp's OEIS Frontend

A386671 All integers k that can produce a closed walk in a square lattice via noncongruent primitive k-length hypotenuses, in ascending order.

%I A386671 #20 Aug 18 2025 16:06:59
%S A386671 40885,69745,98605,127465,146705,175565,214045,233285,242905,262145,
%T A386671 271765,329485,358345,377585,416065,435305,464165,473785,550745,
%U A386671 560365,579605,618085,646945,666185,675805,695045,704665,752765,762385,787205,810485,839345,848965
%N A386671 All integers k that can produce a closed walk in a square lattice via noncongruent primitive k-length hypotenuses, in ascending order.
%C A386671 Most of the initial terms are (5 * 13 * 37) * p, where p is a Pythagorean prime (A002144) other than 5, 13, or 37. These terms produce walks of 8 segments that form the same bilaterally symmetrical base shape, scaled up by a factor of p, and rotated such that the axis of symmetry becomes the hypotenuse of Pythagorean triangle [x, y, p] * 6392. See Links for examples.
%C A386671 Other series with their own such patterns occur with a series based on (29 * 61 * 89) * p beginning at a(30) = 787205, and another based on (13 * 109 * 229) * p beginning at a(60) = 1622465. It is conjectured that there are infinitely many of such series.
%C A386671 The first term that is a product of 5 different Pythagorean primes, a(44) = 1185665, is also the first that produces multiple different solutions.
%C A386671 It is conjectured that all terms are products of 4 or more Pythagorean primes, including at least 4 different ones, though not all such products produce closed walks. It is also conjectured that all walks have an even count of hypotenuse segments and a minimum of 6 segments, and are bilaterally symmetrical when arranged in order of increasing angle.
%C A386671 The shortest such walk in 2 or more dimensions, by total walk length rather than diagonal segment length, gives A385525, with A385525(2) = a(1) * 8. Closed walks along diagonals in an equilateral triangular lattice gives A387031.
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_4.png">Base shape of walk where segment length a(n) is a Pythagorean prime multiple of 5 * 13 * 37</a>
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_5.png">a(1): (5 * 13 * 37) * 17, axis slope 15/8</a>
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_6.png">a(2): (5 * 13 * 37) * 29, axis slope 21/20</a>
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_7.png">a(3): (5 * 13 * 37) * 41, axis slope 40/9</a>
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_8.png">a(4): (5 * 13 * 37) * 53, axis slope 45/28</a>
%H A386671 Charles L. Hohn, <a href="/A386671/a386671_9.png">a(30): (29 * 61 * 89) * 5, axis slope 4/3</a>
%e A386671 a(1) = 40885 because 40885-length hypotenuses of primitive Pythagorean triangles [38076, 14893] + [35844, 19667] + [11603, 39204] + [-34387, 22116] + [-37523, 16236] + [-26093, -31476] + [3636, -40723] + [8844, -39917] = [0, 0].
%o A386671 (PARI) is_a386671(k)={my(v=List); for(x=1, sqrtint(k^2/2), my(y=sqrtint(k^2-x^2)); if(x^2+y^2==k^2 && gcd([x, y, k])==1, listput(v, [x, y]))); return(if(#v>=3 && closable(v), 1, 0))}
%o A386671 closable(v, c=vector(3))={my(o=!c[1] && !c[2]); if(#v==1, return(if(c[3] && (o || vecsort(abs([c[1], c[2]]))==vecsort(abs(v[1]))), 1, 0))); my(x, y, v2=v); listpop(v2); foreach(if(o, [0, 1], [0, 1, -1]), x, foreach(if(!x, [0], o, [1], [1, -1]), y, forperm(2, p, if(closable(v2, c+[v[#v][p[1]]*x, v[#v][p[2]]*y, abs(x)]), return(1)); if(!x || o, break)))); 0}
%Y A386671 Cf. A385525, A387031.
%Y A386671 Cf. A002144, A020882.
%K A386671 nonn
%O A386671 1,1
%A A386671 _Charles L. Hohn_, Jul 28 2025