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A213325 Number of ways to write n = q + sum_{k=1}^m(-1)^{m-k}p_k, where p_k is the k-th prime, and q is a practical number with q-4 and q+4 also practical.

%I A213325 #10 Mar 03 2013 11:49:27
%S A213325 0,0,0,0,0,0,0,0,1,1,1,1,2,1,2,2,2,1,2,3,3,1,2,4,3,2,3,4,4,3,3,4,4,4,
%T A213325 4,4,5,4,4,5,5,3,3,4,4,4,3,4,4,4,3,4,4,4,3,3,5,3,2,4,6,4,3,6,7,2,2,6,
%U A213325 6,2,2,5,7,2,2,5,6,3,3,3,7,3,2,3,7,4,5,4,8,2,5,4,6,2,4,2,5,3,5,4
%N A213325 Number of ways to write n = q + sum_{k=1}^m(-1)^{m-k}p_k, where p_k is the k-th prime, and q is a practical number with q-4 and q+4 also practical.
%C A213325 Conjecture: a(n)>0 for all n>8.
%C A213325 The author has verified this for n up to 5*10^6.
%H A213325 Zhi-Wei Sun, <a href="/A213325/b213325.txt">Table of n, a(n) for n = 1..10000</a>
%H A213325 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;59d09263.1303">More conjectures on alternating sums of consecutive primes</a>, a message to Number Theory List, March 2, 2013.
%e A213325 a(11)=1 since 11=8+(7-5+3-2) with 4, 8, 12 all practical.
%t A213325 f[n_]:=f[n]=FactorInteger[n]
%t A213325 Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t A213325 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 A213325 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t A213325 q[n_]:=q[n]=pr[n-4]==True&&pr[n]==True&&pr[n+4]==True
%t A213325 s[0_]:=0
%t A213325 s[n_]:=s[n]=Prime[n]-s[n-1]
%t A213325 a[n_]:=a[n]=Sum[If[n-s[m]>0&&q[n-s[m]],1,0],{m,1,n}]
%t A213325 Table[a[n],{n,1,100}]
%Y A213325 Cf. A008347, A005153, A208246, A213202.
%K A213325 nonn
%O A213325 1,13
%A A213325 _Zhi-Wei Sun_, Mar 03 2013