A208249 Number of primes p with n < p < 2n, such that p-1 and p+1 are both practical.

0, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 2, 2, 3, 4, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Zhi-Wei Sun, Jan 12 2013

Keywords

• nonn

Comments

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

Zhi-Wei Sun also made the following conjectures:

(1) For each integer n>6 there is a practical number q with n

(2) For any integer n>231 there is a prime p with n

(3) There are infinitely many twin prime pairs {p,p+2} with p-1,p+1,p+3 all practical.

(4) Any odd number n>1 can be written as p+q (p,q>0) with p practical and p^2+q^2 prime.

Examples

			a(9)=1 since 17 is the only prime 9<p<18 with p-1 and p+1 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.

Crossrefs

Cf. A000040, A001097, A005153, A208243, A208244, A208246.

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[n+k] == True && pr[n+k-1] == True && pr[n+k+1] == True,1,0], {k,1,n-1}]; Table[a[n],{n,1,100}]

Formula

a(n) = card { p in A000040 | n < p < 2n, p-1 and p+1 are both practical }.

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