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 A209254 Number of ways to write 2n-1 = p+q with q practical, p and p^4+q^4 both prime. 14
 0, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 4, 3, 1, 3, 1, 1, 4, 2, 5, 5, 1, 4, 1, 2, 4, 3, 1, 6, 3, 4, 4, 5, 1, 6, 7, 2, 4, 3, 4, 2, 4, 5, 1, 2, 3, 7, 5, 2, 4, 8, 4, 6, 5, 1, 2, 2, 3, 8, 3, 1, 5, 6, 2, 4, 7, 4, 8, 4, 2, 7, 6, 3, 4, 3, 1, 6, 6, 1, 7, 6, 2, 8, 9, 5, 7, 3, 3, 10, 7, 3, 9, 14, 1, 9, 4, 3, 4, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Conjecture: a(n)>0 for all n>1. Zhi-Wei Sun also conjectured that any odd integer greater than one can be written as p+q with q practical, and p and p^2+q^2 both prime. This is a refinement of Ming-Zhi Zhang's problem related to A036468. 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)=1 since 2*8-1=11+4 with 4 practical, 11 and 11^4+4^4=14897 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[pr[2n-1-Prime[k]]==True&&PrimeQ[Prime[k]^4+(2n-1-Prime[k])^4]==True, 1, 0], {k, 1, PrimePi[2n-1]}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A000040, A005153, A208243, A208244, A208246, A209253. Sequence in context: A010244 A141298 A256915 * A227738 A103960 A242626 Adjacent sequences:  A209251 A209252 A209253 * A209255 A209256 A209257 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 14 2013 STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)