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%I #5 Aug 04 2015 10:13:24
%S 0,0,0,2,7,0,0,18,4,16,11,2,12,4,35,3,8,20,2,29,7,5,5,6,7,7,7,9,10,12,
%T 2,6,7,13,13,2,10,4,64,30,6,61,12,18,31,3,13,8,52,11,6,13,36,26,10,3,
%U 52,25,57,26,37,6,14,48,16,42,25,2,4,10
%N Least positive integer k < prime(n) such that k + k' is a triangular number, where k' is the unique integer among 1,...,prime(n)-1 with k*k' == 1 (mod prime(n)), or 0 if no such k exists.
%C Conjecture: a(n) > 0 for all n > 7. In other words, for any prime p > 17 there are two integers k and k' among 1,...,p-1 with k*k' == 1 (mod p) such that k + k' is a triangular number.
%C See also A260928 for a similar conjecture involving squares.
%H Zhi-Wei Sun, <a href="/A260929/b260929.txt">Table of n, a(n) for n = 1..10000</a>
%e a(5) = 7 since 7*8 is congruent to 1 modulo prime(5)=11, and 7 + 8 = 5*6/2 is a triangular number.
%e a(8) = 18 since 18*18 is congruent to 1 modulo prime(8)=19, and 18 + 18 = 8*9/2 is a triangular number.
%t TQ[n_]:=IntegerQ[Sqrt[8n+1]]
%t Do[Do[If[TQ[k+PowerMod[k,-1,Prime[n]]],Print[n," ",k];Goto[aa]];Continue,{k,1,Prime[n]-1}];
%t Print[n," ",0];Label[aa];Continue,{n,1,70}]
%Y Cf. A000040, A000217, A260928.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Aug 04 2015
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