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%I #27 Jan 31 2014 03:45:06
%S 0,3,3,5,3,5,7,3,3,5,7,11,11,13,17,17,3,5,7,37,3,5,7,17,11,13,23,17,
%T 19,3,5,7,3,5,7,41,11,13,47,17,19,53,23,31,59,29,3,3,5,7,11,11,13,17,
%U 17,19,23,23,61,29,29,3,5,7,3,5,7,3,5,7,79,11,13,109,17,19,61,23,31,67,29,31,73,41,37,79,3,5,7,47,11,13,3,5,7,59,11,13,3,5
%N Least odd prime p such that 2*n - p = sigma(k) for some k > 0, or 0 if such an odd prime p does not exist, where sigma(k) is the sum of all (positive) divisors of k.
%C Conjecture: a(n) > 0 for all n > 1. Moreover, if n > 180 is not among 284, 293, 371, 542, 788, 1274, then 2*n can be written as p + sigma(m^2), where p is an odd prime and m is a positive integer.
%C See also part (i) of the conjecture in A233654.
%C Note that if sigma(k) is odd, then the order of k at each odd prime must be even, and hence k has the form m^2 or 2*m^2, where m is a positive integer.
%C We have verified part (i) of the conjecture for n up to 10^9.
%H Zhi-Wei Sun, <a href="/A233793/b233793.txt">Table of n, a(n) for n = 1..10000</a>
%e a(2) = 3 since 2*2 = 3 + sigma(1), but 2*2 = 2 + sigma(k) for no k > 0.
%t sigma[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
%t S[n_]:=Union[Table[sigma[j^2],{j,1,Sqrt[n]}],Table[sigma[2*j^2],{j,1,Sqrt[n/2]}]]
%t Do[Do[If[MemberQ[S[2n],2n-Prime[k]],Print[n," ",Prime[k]];Goto[aa]],{k,2,PrimePi[2n]}];
%t Print[n," ",0];Label[aa];Continue,{n,1,100}]
%Y Cf. A000040, A000203, A232270, A233544, A233654.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Dec 15 2013
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