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A214841 Number of ways to write n=p+q/2, where p and q are practical numbers smaller than n. 1

%I #13 Dec 08 2018 11:22:00

%S 0,0,1,0,1,1,2,1,2,2,1,1,1,2,1,1,1,2,2,3,3,4,1,3,2,4,2,4,2,5,3,5,4,6,

%T 2,5,2,6,3,5,2,6,3,7,4,6,1,6,3,6,4,6,1,5,3,6,6,6,2,7,2,6,5,6,2,7,3,8,

%U 6,7,1,8,3,7,6,7,1,7,3,7,8,7,2,9,2,7,7,8,3,9,3,10,8,8,2,11,3,9,8,9

%N Number of ways to write n=p+q/2, where p and q are practical numbers smaller than n.

%C Conjecture: a(n)>0 for all n>4.

%C This has been verified for n up to 5*10^6.

%H Zhi-Wei Sun, <a href="/A214841/b214841.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(23)=1 since 23=20+6/2 with 6 and 20 practical and smaller than 23.

%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[2k]==True&&pr[n-k]==True,1,0],{k,1,(n-1)/2}]

%t Table[a[n],{n,1,100}]

%Y Cf. A005153.

%K nonn

%O 1,7

%A _Zhi-Wei Sun_, Mar 08 2013

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