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%I #17 Dec 05 2018 05:10:56
%S 0,0,1,2,2,2,2,1,2,3,2,4,1,2,3,3,3,4,2,4,3,4,3,6,3,2,3,4,4,6,3,5,3,4,
%T 5,8,3,2,5,5,4,7,4,7,4,2,4,11,3,1,4,7,4,7,6,7,3,4,5,12,3,2,4,8,7,8,5,
%U 9,4,2,6,14,5,2,6,7,7,9,5,9,4,4,5,14,8,2,5,8,7,10,6,9,6,2,8,15,5,3,5,8
%N Number of practical numbers p<n with n-p and n+p both prime or both practical.
%C Conjecture: a(n)>0 for all n>2.
%C This has been verified for n up to 10^7.
%C Except for p=1, all practical numbers are even. Thus, (n-p,n+p) prime is possible only if n is odd, and (n-p,n+p) can be practical only if n is even (except for p=1). - _M. F. Hasler_, Jan 19 2013
%H Zhi-Wei Sun, <a href="/A209312/b209312.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(8)=1 since 4, 8-4 and 8+4 are all practical.
%e a(13)=1 since 6 is practical, and 13-6 and 13+6 are both prime.
%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[pr[p]==True&&((PrimeQ[n-p]==True&&PrimeQ[n+p]==True)||(pr[n-p]==True&&pr[n+p]==True)),1,0],{p,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%o (PARI) A209312(n)=sum(p=1,n-1, is_A005153(p) && ((is_A005153(n-p) && is_A005153(n+p)) || (isprime(n-p) && isprime(n+p)))) \\ (Could be made more efficient by separating the case of odd and even n.) - _M. F. Hasler_, Jan 19 2013
%Y Cf. A005153, A208243, A208244, A208246, A208249, A209253, A209254.
%Y Cf. A209321: Indices for which a(n)=2.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Jan 19 2013
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