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A210681
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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.
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4
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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, 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, 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
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OFFSET
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1,6
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COMMENTS
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Conjecture: a(n) > 0 for all n > 4.
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.
Zhi-Wei Sun also made the following conjectures:
(1) Any even number greater than 10 can be written as the sum of four elements in the set
S = {prime p: p-1 and p+1 are both practical}.
Also, every n=3,4,5,... can be represented as the sum of a prime in S and two triangular numbers.
(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.
(3) Every n=3,4,... can be written as the sum of three elements in the set
T = {x: 6x is practical with 6x-1 and 6x+1 both prime}.
(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.
(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.
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LINKS
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Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv:1211.1588 [math.NT], 2012-2017.)
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EXAMPLE
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a(5)=1 since 2*5=3+3+4 with 3 and 5 both prime, and 2 and 4 both practical.
a(6)=2 since 2*6=3+3+6=3+5+4 with 3,5,7 all prime and 2,4,6 all practical.
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MATHEMATICA
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f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True
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]]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
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CROSSREFS
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Cf. A005153, A068307, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210479, A210528, A210531, A210533, A258838.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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