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 A209320 Number of ways to write 2n = p+q with p and q both prime, p+1 and q-1 both practical. 9
 0, 0, 1, 2, 3, 2, 2, 2, 2, 3, 4, 5, 3, 2, 3, 3, 5, 7, 3, 3, 4, 4, 5, 8, 4, 3, 5, 2, 4, 8, 3, 4, 6, 2, 4, 7, 3, 4, 7, 2, 4, 9, 4, 4, 9, 5, 3, 9, 3, 5, 8, 3, 4, 10, 4, 6, 8, 5, 4, 14, 2, 4, 8, 2, 6, 10, 4, 4, 7, 4, 4, 10, 5, 4, 8, 3, 4, 9, 5, 5, 7, 3, 3, 13, 6, 5, 7, 4, 2, 11, 5, 5, 9, 4, 2, 9, 3, 6, 10, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Conjecture: a(n)>0 for all n>2. As p+q=(p+1)+(q-1), this unifies Goldbach's conjecture and its analog involving practical numbers. The conjecture has been verified for n up to 10^7. 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(8) = 2 since 2*8 = 3+13 = 11+5 with 3, 5, 11, 13 all prime and 3+1, 13-1, 11+1, 5-1 all practical. 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[2n-Prime[k]]==True&&pr[Prime[k]+1]==True&&pr[2n-Prime[k]-1]==True, 1, 0], {k, 1, PrimePi[2n-2]}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A005153, A002372, A045917, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315. Sequence in context: A111497 A220554 A208243 * A097051 A323761 A078832 Adjacent sequences:  A209317 A209318 A209319 * A209321 A209322 A209323 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 19 2013 STATUS approved

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Last modified October 15 18:18 EDT 2019. Contains 328037 sequences. (Running on oeis4.)