A073451 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.

1, 1, 1, 1, 2, 4, 0, 12, 6, 66, 156, 44, 312, 1484, 2672, 6680, 19080, 45024, 168496, 2033271, 724543, 2776536, 24598062, 26849699, 345160845, 4478968678, 5094833662, 14184530127, 29116554754, 125878922175

Offset: 1

Author

T. D. Noe, Aug 02 2002

Keywords

• hard
• more
• nice
• nonn

Comments

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.

Examples

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

Links

• Carlos Rivera, Puzzle 189: Squares and primes in a row, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A072129, A073452.

Programs

Mathematica

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]]
				

Extensions

a(24)-a(30) from Martin Ehrenstein, Jul 19 2023