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%I #16 Dec 08 2018 11:21:22
%S 0,0,1,2,0,4,1,0,2,2,0,4,0,1,4,2,0,6,1,3,2,2,0,5,2,1,3,1,2,11,0,1,4,1,
%T 2,6,0,2,4,3,1,9,2,3,4,2,0,7,1,4,4,5,0,8,4,1,3,3,0,15,0,3,4,4,4,13,2,
%U 4,2,5,2,10,0,2,11,2,3,12,0,6,6,2,2,13,3,5,7,5,1,16,4,4,6,3,2,11,0,8,6,7
%N a(n) = |{0<k<n: k*n is practical, k*n-1 and k*n+1 are twin primes}|.
%C Conjecture: a(n)>0 for all n>911.
%C This implies that for each n=2,3,4,... there is a positive integer k<n with k*n practical.
%C The conjecture has been verified for n up to 10^6.
%H Zhi-Wei Sun, <a href="/A210444/b210444.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.
%e a(7) = 1 since 6*7 = 42 is practical, and 41 and 43 are twin primes.
%t f[n_]:=f[n]=FactorInteger[n]
%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t 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}]
%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t a[n_]:=a[n]=Sum[If[PrimeQ[k*n-1]==True&&PrimeQ[k*n+1]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A071558, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A219185, A219312, A219315, A219320.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Jan 20 2013
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