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%I #18 Dec 05 2018 05:11:11
%S 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,
%T 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,
%U 3,3,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2
%N Number of primes p with n < p < 2n, such that p-1 and p+1 are both practical.
%C Conjecture: a(n)>0 for all n>8.
%C Zhi-Wei Sun also made the following conjectures:
%C (1) For each integer n>6 there is a practical number q with n<q<2n such that q-1 and q+1 are both prime.
%C (2) For any integer n>231 there is a prime p with n<p<2n-1 such that p+2 is prime, and p-1 and p+1 are all practical.
%C (3) There are infinitely many twin prime pairs {p,p+2} with p-1,p+1,p+3 all practical.
%C (4) Any odd number n>1 can be written as p+q (p,q>0) with p practical and p^2+q^2 prime.
%H Zhi-Wei Sun, <a href="/A208249/b208249.txt">Table of n, a(n) for n = 1..10000</a>
%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arxiv:1211.1588 [math.NT], 2012-2017.
%F a(n) = card { p in A000040 | n < p < 2n, p-1 and p+1 are both practical }.
%e a(9)=1 since 17 is the only prime 9<p<18 with p-1 and p+1 both practical.
%t 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}]
%Y Cf. A000040, A001097, A005153, A208243, A208244, A208246.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Jan 12 2013
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