OFFSET
1,13
COMMENTS
Conjecture: a(n)>0 for all n>8.
Zhi-Wei Sun also made some similar conjectures, below are few examples.
(1) Each integer n>3 can be written as p+q with p prime or practical, and q and q+2 both practical.
(2) Any integer n>12 can be written as p+q with p prime or practical, and q-8, q, q+8 all practical.
(3) The interval [n,2n) contains a practical number p with p-n a triangular number.
(4) Any integer n>1 can be written as x^2+y (x,y>0) with 2x and 2xy both practical.
Note that if x>=y>0 with x practical then xy is also practical.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures on representations involving primes, arxiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(11)=1 since 11=3+8 with 3 prime, and 4, 8, 12 all practical.
a(12)=1 since 12=4+8 with 4, 8, 12 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[pr[k]==True&&pr[k-4]==True&&pr[k+4]==True&&(PrimeQ[n-k]==True||pr[n-k]==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