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 A209253 Number of ways to write 2n-1 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 (list; graph; refs; listen; history; text; internal format)
 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: A194310 A306227 A272231 * A165113 A069903 A331003 Adjacent sequences:  A209250 A209251 A209252 * A209254 A209255 A209256 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 14 2013 STATUS approved

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Last modified December 1 00:44 EST 2020. Contains 338831 sequences. (Running on oeis4.)