OFFSET
1,4
COMMENTS
Conjecture: a(n)>0 for all n=3,4,...
The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
Zhi-Wei Sun also made the following similar conjectures:
(1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
(2) Each integer n>10 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+6 both practical.
Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+2 both 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(14)=2 since 2*14-1=27=11+16=23+4, where 11 and 23 are primes, 16,16+2,4,4+2 are practical 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)
a[n_]:=a[n]=Sum[If[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n-1-2k]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 11 2013
STATUS
approved