A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 19 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>8.

This has been verified for n up to 10^7.

As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927.

Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime, one of q and q+1 is prime and another of q and q+1 is practical.

Examples

			a(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime.
		

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.

Crossrefs

Cf. A005153, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185.

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[PrimeQ[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True,1,0],{p,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]