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%I #19 Dec 08 2018 11:21:39
%S 0,0,0,0,0,1,1,3,3,4,5,3,5,3,4,4,3,4,4,4,6,6,8,6,8,3,7,3,6,5,5,5,7,6,
%T 11,8,12,4,8,4,7,8,6,8,8,7,11,9,13,5,8,4,7,7,6,6,6,5,7,6,10,4,9,3,9,7,
%U 8,7,6,6,7,4,7,4,7,4,8,8,11,7,6,6,8,5,6,4,7,2,9,7,12,8,7,4,10,5,9,5,8,5
%N Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number.
%C Conjecture: a(n)>0 for all n>5.
%C This has been verified for n up to 300000.
%C Note that for n=406 we cannot represent n in the given way with q+1 practical.
%H Zhi-Wei Sun, <a href="/A211165/b211165.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://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;889694d7.1301">New Goldbach-type conjectures involving primes and practical numbers</a>, a message to Number Theory List, Jan. 29, 2013.
%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(6)=a(7)=1 since 6=3+3+0 and 7=3+3+1 with 3 and 5 both prime, 2 and 4 both practical, 0 and 1 Fibonacci numbers.
%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 pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
%t q[n_]:=q[n]=PrimeQ[n]==True&&PrimeQ[n+2]==True
%t a[n_]:=a[n]=Sum[If[k!=2&&Fibonacci[k]<n&&pp[j]==True&&q[n-Fibonacci[k]-Prime[j]]==True,1,0],{k,0,2*Log[2,n]},{j,1,PrimePi[n-Fibonacci[k]]}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A000045, A001359, A210479, A210681, A210722, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Jan 30 2013
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