A208246 Number of ways to write n = p+q with p prime or practical, and q-4, q, q+4 all practical.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 4, 4, 3, 4, 5, 4, 3, 5, 6, 4, 3, 5, 7, 5, 4, 6, 8, 4, 3, 5, 8, 4, 2, 4, 8, 5, 3, 4, 7, 4, 3, 5, 7, 3, 2, 4, 6, 5, 4, 4, 7, 5, 4, 5, 7, 4, 2, 4, 7, 5, 3, 4, 6, 4, 4, 6, 6, 3, 2, 5, 6, 4, 4, 5, 7, 5, 5, 7, 8, 2, 2, 6, 8, 5, 3, 4, 7

Offset: 1

Author

Zhi-Wei Sun, Jan 11 2013

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A005153, A208243, A208244, A219842, A220272.

Programs

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}]