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

1,2

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.

Links

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

Example

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

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}]

Crossrefs

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

Sequence in context: A182731 A267518 A101772 * A158894 A352445 A131919

Adjacent sequences: A233790 A233791 A233792 * A233794 A233795 A233796

Keyword

nonn

Author

Zhi-Wei Sun, Dec 15 2013

Status

approved