A387031 All integers k that can produce a closed walk in an equilateral triangular lattice via noncongruent primitive k-length diagonals, in ascending order.

53599, 104377, 105469, 121303, 126217, 136591, 144781, 172081, 177289, 178087, 189007, 205933, 211603, 222859, 251503, 273637, 276241, 290563, 300181, 300979, 307489, 325717, 345247, 346801, 348859, 358267, 359233, 388759, 392119, 392977, 403039, 417487

Offset: 1

Author

Charles L. Hohn, Aug 13 2025

Keywords

• nonn

Comments

All observed terms are products of 4 different primes that are 1 mod 6 (A002476), though not all such products produce closed walks. It is conjectured that all terms are products of 4 or more such primes, including at least 4 different ones.

Closed walks along diagonals in a square lattice gives A386671.

Examples

			a(1) = 53599 because 53599-length moves of [0, 60, 120] degrees respectively of [26216, 35445, 0] + [-16165, 0, 43656] + [-19651, 0, 41000] + [-1389, -52891, 0] + [0, -21829, -39240] + [50264, 0, -6141] = [39275, -39275, 39275], and [0, 0] + 39275 @ 0 degrees - 39275 @ 60 degrees + 39275 @ 120 degrees = [0, 0].
		

Crossrefs

Cf. A002476, A386671.

Programs

PARI

is_a387031(k)={my(v=List); for(x=1, sqrtint(k^2/3), my(y=(sqrtint(4*k^2-3*x^2)-x)/2); if((x+y/2)^2+y^2*3/4==k^2 && gcd([x, y, k])==1, listput(v, [x, y]))); return(if(#v>=3 && closable(v), 1, 0))}
closable(v, c=vector(4))={my(o=(c[1]==c[3] && c[2]==-c[3])); if(!#v, return(if(c[4], o, 0))); my(x, y, v2=v); listpop(v2); foreach(if(o, [0, 1], [0, 1, 2, 3, 4, 5, 6]), r1, my(r2=r1%6+1); forperm(2, p, my(c2=c); if(r1, c2[(r1-1)%3+1]+=v[#v][p[1]]*if(r1>=4, -1, 1); c2[(r2-1)%3+1]+=v[#v][p[2]]*if(r2>=4, -1, 1); c2[4]++); if(closable(v2, c2), return(1)); if(!r1, break)))}