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A210533
Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.
4
0, 0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 1, 5, 3, 5, 5, 5, 3, 6, 5, 6, 6, 6, 3, 6, 2, 7, 7, 6, 6, 7, 1, 6, 8, 8, 3, 8, 4, 6, 8, 7, 4, 8, 4, 8, 8, 6, 5, 8, 5, 6, 9, 7, 3, 9, 6, 8, 9, 8, 5, 9, 3, 7, 9, 7, 5, 9, 2, 7, 9, 7, 4, 10, 4, 8, 10, 8, 5, 10, 8, 7, 10, 10, 6, 10, 4, 9, 11, 8, 7, 11, 6, 11, 12, 11
OFFSET
1,4
COMMENTS
Conjecture: a(n)>0 for all n>2. Moreover, for each m=2,3,4,... any sufficiently large even integer can be written as x+y (x,y>0) with x-1 and x+1 both prime, and x and x^m+y^m both practical.
LINKS
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(17)=1 since 2*17=12+22 with 11 and 13 both prime, and 12 and 12^3+22^3=12376 both practical.
MATHEMATICA
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)
a[n_]:=a[n]=Sum[If[PrimeQ[2k-1]==True&&PrimeQ[2k+1]==True&&pr[2k]==True&&pr[(2k)^3+(2n-2k)^3]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 28 2013
STATUS
approved