cp's OEIS Frontend

A227456 Number of permutations i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = 1 such that all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are of the form (p+1)/4 with p a prime congruent to 3 modulo 4.

%I A227456 #19 Aug 04 2019 08:46:22
%S A227456 1,1,1,1,1,2,4,11,15,15
%N A227456 Number of permutations i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = 1 such that all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are of the form (p+1)/4 with p a prime congruent to 3 modulo 4.
%C A227456 Conjecture: a(n) > 0 for all n > 0. Similarly, for any positive integer n, there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are of the form (p-1)/4 with p a prime congruent to 1 modulo 4.
%C A227456 Note that if a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n  with i_0 = 0 makes all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 of the form (p+1)/4 with p a prime, then we must have i_n = 1. This can be explained as follows: If i_n > 1, then 3 | i_n since 4*(i_n^2+0)-1 is a prime not divisible by 3, and similarly i_{n-1},...,i_1 are also multiples of 3 since 4*(i_{n-1}^2+i_n)-1, ..., 4*(i_1^2+i_2)-1 are primes not divisible by 3. Therefore, i_n > 1 would lead to a contradiction.
%H A227456 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1309.1679">Some new problems in additive combinatorics</a>, preprint, arXiv:1309.1679 [math.NT], 2013-2014.
%e A227456 a(1) = a(2) = a(3) = a(4) = a(5) = 1 due to the permutations (0,1), (0,2,1), (0,3,2,1), (0,3,2,4,1), (0,3,2,4,5,1).
%e A227456 a(6) = 2 due to the permutations
%e A227456     (0,3,6,2,4,5,1) and (0,3,6,5,2,4,1).
%e A227456 a(7) = 4 due to the permutations
%e A227456     (0,3,6,2,4,5,7,1), (0,3,6,2,7,4,5,1),
%e A227456     (0,3,6,5,2,7,4,1), (0,3,6,5,7,4,2,1).
%e A227456 a(8) = 11 due to the permutations
%e A227456 (0,3,6,2,4,5,8,7,1), (0,3,6,2,7,8,4,5,1), (0,3,6,2,8,4,5,7,1),
%e A227456 (0,3,6,2,8,7,4,5,1), (0,3,6,5,2,7,8,4,1), (0,3,6,5,2,8,7,4,1),
%e A227456 (0,3,6,5,7,8,2,4,1), (0,3,6,5,7,8,4,2,1), (0,3,6,5,8,2,7,4,1),
%e A227456 (0,3,6,5,8,4,2,7,1), (0,3,6,5,8,7,4,2,1).
%e A227456 a(9) > 0 due to the permutation (0,3,6,9,2,4,5,8,7,1).
%e A227456 a(10) > 0 due to the permutation (0,3,6,9,2,4,5,10,8,7,1).
%t A227456 (* A program to compute required permutations for n = 8. *)
%t A227456 f[i_,j_]:=f[i,j]=PrimeQ[4(i^2+j)-1]
%t A227456 V[i_]:=V[i]=Part[Permutations[{2,3,4,5,6,7,8}],i]
%t A227456 m=0
%t A227456 Do[Do[If[f[If[j==0,0,Part[V[i],j]],If[j<7,Part[V[i],j+1],1]]==False,Goto[aa]],{j,0,7}];
%t A227456 m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]," ",1];Label[aa];Continue,{i,1,7!}]
%Y A227456 Cf. A229082, A229141, A229130.
%K A227456 nonn,more,hard
%O A227456 1,6
%A A227456 _Zhi-Wei Sun_, Sep 16 2013