A330008 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to an unvisited square with the lowest prime spiral number, or lowest composite number if no primes are available.

1, 10, 3, 6, 17, 4, 7, 2, 5, 8, 11, 14, 29, 32, 61, 16, 19, 22, 41, 18, 37, 62, 139, 60, 13, 28, 9, 12, 31, 54, 89, 30, 53, 26, 47, 76, 23, 20, 43, 70, 109, 42, 73, 44, 71, 40, 67, 36, 97, 34, 59, 56, 131, 88, 127, 52, 83, 80, 167, 82, 173, 84, 27, 24, 79, 46, 21, 72, 107

Offset: 1

Author

Simon S. Gurvets, Nov 26 2019

Keywords

• nonn
• fini
• full

Comments

The squares are numbered starting with 1 at the origin (0,0). The sequence is finite: when arriving on square number a(209) = 147, there is no free square within reach for the next move. - M. F. Hasler, Jan 26 2020

Links

• Simon S. Gurvets, Table of n, a(n) for n = 1..209
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).
• Scott R. Shannon, Image showing the steps of the knight's path. A green dot marks the starting 1 square and a red dot the final square with number 147. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 15. Purple dots mark the visited squares containing a prime number. The path after square 1 contains 67 primes and 141 composites.

Crossrefs

Cf. A316667.

Programs

PARI

local(U); my(v(p)=if(bittest(U,p),[9,0],[1-isprime(p+1),p]), nxt(x)=vecsort([v(pos(x+k))|k<-K])[1][2], K=[[(-1)^(i\2)<<(i>4),(-1)^i<<(i<5)]|i<-[1..8]], pos(x,y=x[2])=if(y>=abs(x=x[1]),4*y^2-y-x,-x>=abs(y),4*x^2-x-y,-y>=abs(x),(4*y-3)*y+x,(4*x-3)*x+y), xy(n,m=sqrtint(n),k=m\/2)=if(m<=n-=4*k^2,[n-3*k,-k],n>=0,[-k,k-n],n>=-m,[-k-n,k],[k,3*k+n]), A=List(0)); until(!listput(A,nxt(xy(A[#A]))), U+=1<A330008=[t+1|t<-A[^-1]] \\ M. F. Hasler, Jan 26 2020