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%I #7 May 25 2014 09:34:37
%S 0,0,0,2,2,2,3,2,2,3,4,2,2,3,2,3,4,7,2,2,3,4,2,3,4,13,6,7,11,13,10,11,
%T 2,3,4,18,6,7,2,3,4,2,2,3,4,6,6,2,3,2,2,3,4,2,2,3,4,6,6,2,3,2,3,4,7,2,
%U 3,2,3,4,7,2,2,2,2,3,2,3,4,7
%N Least positive integer k < n such that k*p == 1 (mod prime(k)) for some prime p < prime(k) and (n-k)*q == 1 (mod prime(n-k)) for some prime q < prime(n-k), or 0 if such a number k does not exist.
%C According to the conjecture in A242753, a(n) should be positive for all n > 3.
%C We have verified that a(n) > 0 for all n = 4, ..., 10^8.
%H Zhi-Wei Sun, <a href="/A242879/b242879.txt">Table of n, a(n) for n = 1..10000</a>
%e a(4) = 2 since 4 = 2 + 2 and 2*2 == 1 (mod prime(2)=3).
%e a(7) = 3 since 7 = 3 + 4, 3*2 == 1 (mod prime(3)=5) with 2 prime, and also 4*2 == 1 (mod prime(4)=7) with 2 prime, but 5*9 == 1 (mod prime(5)=11) with 9 not prime.
%t p[n_]:=PrimeQ[PowerMod[n,-1,Prime[n]]]
%t Do[Do[If[p[k]&&p[n-k],Print[n," ",k];Goto[aa]];Continue,{k,1,n/2}];Print[n," ",0];Label[aa];Continue,{n,1,80}]
%Y Cf. A000040, A242425, A242748, A242753, A242754, A242755.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, May 25 2014
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