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A073452 a(n) is the number of essentially different ways in which the integers 1,2,3,...,n can be arranged in a sequence such that all pairs of adjacent integers sum to a prime number. Rotations and reversals are counted only once.

%I A073452 #11 Jul 19 2023 07:30:45
%S A073452 1,1,1,1,2,3,12,16,70,232,1072,3136,11648,18388,95772,452136,2047488,
%T A073452 5565488,22802028,60841609,337801784,2116714332,11425028900,
%U A073452 69023494710,429917269469
%N A073452 a(n) is the number of essentially different ways in which the integers 1,2,3,...,n can be arranged in a sequence such that all pairs of adjacent integers sum to a prime number. Rotations and reversals are counted only once.
%C A073452 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=6 exhibits rotational solutions: {1,4,3,2,5,6}, which is actually a prime circle. See A051252 for more details about prime circles.
%C A073452 The Mathematica program uses a backtracking algorithm to count the arrangements. To print the unique arrangements, remove the comments from around the print statement. It seems that a greedy algorithm can be used to quickly find a solution for any n.
%H A073452 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_020.htm">Conjecture 20. The first N natural numbers listed in an order such that the sum of each two adjacent of them is a prime number, and the Rivera's Algorithm</a>, The Prime Puzzles & Problems Connection.
%H A073452 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 A073452 a(5)=2 because there are two essential different arrangements: {1,4,3,2,5} and {3,4,1,2,5}.
%t A073452 nMax=12; $RecursionLimit=500; try[lev_] := Module[{t, j, circular}, If[lev>n, circular=PrimeQ[soln[[1]]+soln[[n]]]; If[(!circular&&soln[[1]]<soln[[n]])||(circular&&soln[[1]]==1&&soln[[2]]<=soln[[n]]), (*Print[soln]; *)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+j], 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 A073452 Cf. A073451, A051252.
%K A073452 hard,more,nice,nonn
%O A073452 1,5
%A A073452 _T. D. Noe_, Aug 02 2002
%E A073452 a(21)-a(25) from _Martin Ehrenstein_, Jul 19 2023