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]<n&&pp[j]==True&&q[n-Fibonacci[k]-Prime[j]]==True, 1, 0], {k, 0, 2*Log[2, n]}, {j, 1, PrimePi[n-Fibonacci[k]]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 30 2013
STATUS
approved