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%I #14 Dec 08 2018 11:21:35
%S 0,0,0,0,0,0,0,0,1,0,3,1,4,2,4,2,4,3,4,4,3,4,5,3,8,5,8,4,5,4,5,3,5,4,
%T 4,3,9,2,12,4,9,5,7,6,7,5,5,7,10,5,13,6,10,6,8,6,7,5,1,7,7,1,10,5,8,4,
%U 9,7,8,6,3,10,6,6,10,7,7,9,11,7,10,10,5,10,7,5,10,7,4,8,8,5,11,5,8,10,7,5,12,5
%N Number of ways to write n = (2-(n mod 2))p+q+2^k with p, q-1, q+1 all prime, and p-1, p+1, q all practical.
%C Conjecture: a(n)>0 except for n = 1,...,8, 10, 520, 689, 740.
%C Zhi-Wei Sun also guessed that any integer n>6 different from 407 can be written as p+q+F_k, where p is a prime with p-1 and p+1 practical, q is a practical number with q-1 and q+1 prime, and F_k (k>=0) is a Fibonacci number.
%H Zhi-Wei Sun, <a href="/A210722/b210722.txt">Table of n, a(n) for n = 1..20000</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;20e70044.1301">Sandwiches with primes and practical numbers</a>, a message to Number Theory List, Jan. 13, 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(1832)=1 since 1832=2*881+6+2^6 with 5, 7, 881 all prime and 6, 880, 882 all practical.
%e a(11969)=1 since 11969=127+11778+2^6 with 127, 11777, 11779 all prime and 126, 128, 11778 all practical.
%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 pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True
%t a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[n-2^k-(2-Mod[n,2])Prime[j]]==True,1,0],{k,0,Log[2,n]},{j,1,PrimePi[(n-2^k)/(2-Mod[n,2])]}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533, A210681.
%K nonn
%O 1,11
%A _Zhi-Wei Sun_, Jan 29 2013
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