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%I #49 Dec 17 2017 10:06:29
%S 0,0,0,0,1,2,3,3,3,2,2,3,5,6,4,3,2,4,7,10,8,5,3,5,8,12,10,6,2,3,6,11,
%T 12,6,4,3,5,9,10,6,5,4,5,8,8,5,7,7,6,8,7,6,6,8,6,7,8,5,7,8,6,7,7,4,6,
%U 7,5,6,8,4,8,6,4,5,7,5,5,8,5,6,8,6,4,7,6,6,7,5,3,7,3,4,8,6,8,5,4,3,7,6
%N Number of ways to write 2n = p+q+r (p<=q) with p, q, r-1, r+1 all prime and p-1, p+1, q-1, q+1, r all practical.
%C Conjecture: a(n) > 0 for all n > 4.
%C This conjecture involves two kinds of sandwiches introduced by the author, and it is much stronger than the Goldbach conjecture for odd numbers. We have verified the conjecture for n up to 10^7.
%C Zhi-Wei Sun also made the following conjectures:
%C (1) Any even number greater than 10 can be written as the sum of four elements in the set
%C S = {prime p: p-1 and p+1 are both practical}.
%C Also, every n=3,4,5,... can be represented as the sum of a prime in S and two triangular numbers.
%C (2) Each integer n>7 can be written as p + q + x^2 (or p + q + x(x+1)/2), where p is a prime with p-1 and p+1 both practical, and q is a practical number with q-1 and q+1 both prime.
%C (3) Every n=3,4,... can be written as the sum of three elements in the set
%C T = {x: 6x is practical with 6x-1 and 6x+1 both prime}.
%C (4) Any integer n>6 can be represented as the sum of two elements of the set S and one element of the set T.
%C (5) Each odd number greater than 11 can be written in the form 2p+q+r, where p and q belong to S, and r is a practical number with r-1 and r+1 both prime.
%H Zhi-Wei Sun, <a href="/A210681/b210681.txt">Table of n, a(n) for n = 1..10000</a>
%H G. Melfi, <a href="https://doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56(1996), 205-210.
%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://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(5)=1 since 2*5=3+3+4 with 3 and 5 both prime, and 2 and 4 both practical.
%e a(6)=2 since 2*6=3+3+6=3+5+4 with 3,5,7 all prime and 2,4,6 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&&pp[k]==True&&pq[2n-Prime[j]-Prime[k]]==True,1,0],{j,1,PrimePi[n-1]},{k,j,PrimePi[2n-Prime[j]]}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A005153, A068307, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210479, A210528, A210531, A210533, A258838.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Jan 29 2013
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