A233793 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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Dec 15 2013

Keywords

• nonn

Comments

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.

See also part (i) of the conjecture in A233654.

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.

We have verified part (i) of the conjecture for n up to 10^9.

Examples

			a(2) = 3 since 2*2 = 3 + sigma(1), but 2*2 = 2 + sigma(k) for no k > 0.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000203, A232270, A233544, A233654.

Programs

Mathematica

sigma[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
S[n_]:=Union[Table[sigma[j^2],{j,1,Sqrt[n]}],Table[sigma[2*j^2],{j,1,Sqrt[n/2]}]]
Do[Do[If[MemberQ[S[2n],2n-Prime[k]],Print[n," ",Prime[k]];Goto[aa]],{k,2,PrimePi[2n]}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]