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 A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime. 9
 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 3, 4, 2, 2, 2, 3, 4, 3, 1, 3, 3, 1, 4, 5, 3, 3, 3, 2, 5, 4, 1, 3, 5, 2, 5, 4, 3, 4, 5, 2, 5, 5, 2, 4, 5, 3, 6, 5, 5, 5, 2, 3, 6, 5, 2, 3, 4, 3, 6, 5, 4, 4, 4, 5, 6, 6, 4, 5, 4, 3, 6, 8, 2, 2, 5, 6, 7, 6, 2, 6, 2, 4, 7, 6, 4, 3, 6, 3, 5, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS Conjecture: a(n)>0 for all n>8. This has been verified for n up to 10^7. As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927. Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime,  one of q and q+1 is prime and another of q and q+1 is practical. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106]. Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017. EXAMPLE a(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime. 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[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True, 1, 0], {p, 1, n-1}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A005153, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185. Sequence in context: A072504 A072499 A060272 * A174713 A129985 A085243 Adjacent sequences:  A209312 A209313 A209314 * A209316 A209317 A209318 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 19 2013 STATUS approved

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Last modified October 20 02:52 EDT 2019. Contains 328244 sequences. (Running on oeis4.)