A209253 Number of ways to write 2n-1 as the sum of a Sophie Germain prime and a practical number.

0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 5, 2, 3, 4, 4, 4, 5, 2, 3, 5, 2, 4, 7, 4, 2, 6, 2, 5, 6, 2, 2, 6, 1, 3, 7, 4, 3, 7, 4, 5, 8, 2, 3, 8, 3, 3, 8, 4, 4, 7, 4, 5, 8, 3, 4, 7, 1, 5, 9, 5, 3, 9, 3, 4, 8, 4, 3, 9, 3, 5, 8, 2, 2, 9, 4, 3, 8, 4, 4, 10, 1, 3, 10, 5, 4, 10, 4, 3, 9, 5, 5, 10, 4

Offset

1,4

Comments

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 5*10^6.

Links

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017

Example

a(40)=1 since 2*40-1=23+56 with 23 a Sophie Germain prime and 56 a practical number.

Mathematica

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k]+1]==True&&pr[2n-1-Prime[k]]==True, 1, 0], {k, 1, PrimePi[2n-1]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A005384, A005153, A208243, A208244, A208246, A208249.

Sequence in context: A359455 A363853 A272231 * A165113 A069903 A331003

Adjacent sequences: A209250 A209251 A209252 * A209254 A209255 A209256

Keyword

nonn

Author

Zhi-Wei Sun, Jan 14 2013

Status

approved