cp's OEIS Frontend

A319480 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that R(i^2,prime(n)) > R(j^2,prime(n)), where R(k,p) (with p an odd prime and k an integer) denotes the unique integer r among 0,1,...,(p-1)/2 for which k is congruent to r or -r modulo p.

%I A319480 #22 Sep 20 2018 06:59:58
%S A319480 0,0,1,3,7,10,14,19,41,42,74,79,85,100,154,163,207,224,245,309,318,
%T A319480 342,449,536,590,553,581,715,738,856,912,1085,1037,1324,1229,1477,
%U A319480 1442,1491,1785,1730,1952,1986,2240,2316,2191,2474,2748,2836,3176
%N A319480 Number of ordered pairs (i,j) with 0 < i < j < prime(n)/2 such that R(i^2,prime(n)) > R(j^2,prime(n)), where R(k,p) (with p an odd prime and k an integer) denotes the unique integer r among 0,1,...,(p-1)/2 for which k is congruent to r or -r modulo p.
%C A319480 Conjecture: Let p be any odd prime and let N(p) be the number of ordered pairs (i,j) with 0 < i < j < p/2 and R(i^2,p) > R(j^2,p). Then N(p) == floor((p+1)/8) (mod 2).
%C A319480 See also A319311 for a similar conjecture.
%H A319480 Zhi-Wei Sun, <a href="/A319480/b319480.txt">Table of n, a(n) for n = 2..1000</a>
%e A319480 a(3) = 0 since prime(3) = 5 and R(1^2,5) = 1 = R(2^2,5).
%e A319480 a(4) = 1 since prime(4) = 7, R(1^2,7) = 1 < R(2^2,7) = 3, R(1^2,7) < R(3^2,7) = 2, and R(2^2,7) = 3 > R(3^2,7) = 2.
%t A319480 R[k_,p_]:=R[k,p]=Abs[Mod[k,p,-p/2]];
%t A319480 t[p_]:=t[p]=Sum[Boole[R[i^2,p]>R[j^2,p]],{j,2,(p-1)/2},{i,1,j-1}]; Table[t[Prime[n]],{n,2,50}]
%Y A319480 Cf. A000040, A000290, A319311.
%K A319480 nonn
%O A319480 2,4
%A A319480 _Zhi-Wei Sun_, Sep 20 2018