A073451 - OEIS

%I #18 Jul 19 2023 07:47:18

%S 1,1,1,1,2,4,0,12,6,66,156,44,312,1484,2672,6680,19080,45024,168496,

%T 2033271,724543,2776536,24598062,26849699,345160845,4478968678,

%U 5094833662,14184530127,29116554754,125878922175

%N Number of essentially different ways in which the squares 1,4,9,...,n^2 can be arranged in a sequence such that all pairs of adjacent squares sum to a prime number. Rotations and reversals are counted only once.

%C Note that when the first and last numbers of an arrangement sum to a prime, then there are n rotations that are treated as one arrangement. The case n=10 exhibits the first of these rotational solutions: {1,4,9,64,49,100,81,16,25,36}. The Mathematica program uses a backtracking algorithm to count the arrangements. To print the unique arrangements, remove the comments from around the print statement.

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_189.htm">Puzzle 189: Squares and primes in a row</a>, The Prime Puzzles & Problems Connection.

%e a(5)=2 because there are two essential different arrangements: {9,4,1,16,25} and {9,4,25,16,1}.

%t nMax=12; $RecursionLimit=500; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=PrimeQ[soln[[1]]^2+soln[[n]]^2]; If[(!circular&&soln[[1]]<soln[[n]])||(circular&&soln[[1]]==1&&soln[[2]]<=soln[[n]]), (*Print[soln^2]; *)cnt++ ], (*else append another number to the soln list*) t=soln[[lev-1]]; For[j=1, j<=Length[s[[t]]], j++, If[ !MemberQ[soln, s[[t]][[j]]], soln[[lev]]=s[[t]][[j]]; try[lev+1]; soln[[lev]]=0]]]]; For[lst={1}; n=2, n<=nMax, n++, s=Table[{}, {n}]; For[i=1, i<=n, i++, For[j=1, j<=n, j++, If[i!=j&&PrimeQ[i^2+j^2], AppendTo[s[[i]], j]]]]; soln=Table[0, {n}]; For[cnt=0; i=1, i<=n, i++, soln[[1]]=i; try[2]]; AppendTo[lst, cnt]]; lst

%Y Cf. A072129, A073452.

%K hard,more,nice,nonn

%O 1,5

%A _T. D. Noe_, Aug 02 2002

%E a(24)-a(30) from _Martin Ehrenstein_, Jul 19 2023