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%I #7 Aug 04 2015 08:23:07
%S 0,0,0,0,0,1,2,3,0,0,4,1,1,0,2,0,3,1,1,1,3,0,2,1,1,2,1,2,3,5,4,1,10,5,
%T 10,2,8,3,1,1,7,2,7,4,2,8,6,3,3,1,8,6,2,1,6,5,6,2,2,5,5,7,7,5,6,5,10,
%U 4,7,5
%N Number of positive integers k < prime(n)/2 such that k + k' is a square, where k' is the unique integer among 1, ..., prime(n)-1 such that k*k' == 1 (mod prime(n)).
%C Conjecture: a(n) > 0 for all n > 22. In other words, for any prime p > 80 there is a positive integer k < p/2 such that k + k' is a square, where k' is the unique integer among 1,...,p-1 with k*k' == 1 (mod p).
%H Zhi-Wei Sun, <a href="/A260928/b260928.txt">Table of n, a(n) for n = 1..10000</a>
%e a(54) = 1 since 38*218 is congruent to 1 modulo prime(54)=251 with 38 < 251/2, and 38 + 218 = 16^2 is a square.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t Do[m=0;Do[If[SQ[k+PowerMod[k,-1,Prime[n]]],m=m+1];Continue,{k,1,(Prime[n]-1)/2}];
%t Print[n," ",m];Continue,{n,1,70}]
%Y Cf. A000040, A000290.
%K nonn
%O 1,7
%A _Zhi-Wei Sun_, Aug 04 2015
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