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 A211165 Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number. 2
 0, 0, 0, 0, 0, 1, 1, 3, 3, 4, 5, 3, 5, 3, 4, 4, 3, 4, 4, 4, 6, 6, 8, 6, 8, 3, 7, 3, 6, 5, 5, 5, 7, 6, 11, 8, 12, 4, 8, 4, 7, 8, 6, 8, 8, 7, 11, 9, 13, 5, 8, 4, 7, 7, 6, 6, 6, 5, 7, 6, 10, 4, 9, 3, 9, 7, 8, 7, 6, 6, 7, 4, 7, 4, 7, 4, 8, 8, 11, 7, 6, 6, 8, 5, 6, 4, 7, 2, 9, 7, 12, 8, 7, 4, 10, 5, 9, 5, 8, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Conjecture: a(n)>0 for all n>5. This has been verified for n up to 300000. Note that for n=406 we cannot represent n in the given way with q+1 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, New Goldbach-type conjectures involving primes and practical numbers, a message to Number Theory List, Jan. 29, 2013. Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017. EXAMPLE a(6)=a(7)=1 since 6=3+3+0 and 7=3+3+1 with 3 and 5 both prime, 2 and 4 both practical, 0 and 1 Fibonacci numbers. 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) pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True q[n_]:=q[n]=PrimeQ[n]==True&&PrimeQ[n+2]==True a[n_]:=a[n]=Sum[If[k!=2&&Fibonacci[k]

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Last modified September 18 03:09 EDT 2021. Contains 347504 sequences. (Running on oeis4.)