

A209253


Number of ways to write 2n1 as the sum of a Sophie Germain prime and a practical number.


17



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
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OFFSET

1,4


COMMENTS

Conjecture: a(n)>0 for all n>1.
This has been verified for n up to 5*10^6.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 20122017


EXAMPLE

a(40)=1 since 2*401=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<3Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k]+1]==True&&pr[2n1Prime[k]]==True, 1, 0], {k, 1, PrimePi[2n1]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A005384, A005153, A208243, A208244, A208246, A208249.
Sequence in context: A194310 A306227 A272231 * A165113 A069903 A331003
Adjacent sequences: A209250 A209251 A209252 * A209254 A209255 A209256


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 14 2013


STATUS

approved



