

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.


4



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
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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



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], 2nPrime[k]], Print[n, " ", Prime[k]]; Goto[aa]], {k, 2, PrimePi[2n]}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



