

A208243


Number of ways to write 2n1 = p+q, where p is a prime, and both q and q+2 are practical numbers (A005153).


20



0, 0, 1, 2, 3, 2, 2, 2, 2, 3, 4, 4, 3, 2, 3, 3, 5, 6, 3, 3, 4, 4, 5, 7, 4, 3, 4, 2, 5, 7, 4, 4, 5, 4, 5, 7, 4, 5, 8, 2, 5, 7, 5, 5, 6, 6, 4, 7, 4, 5, 9, 3, 5, 9, 4, 6, 6, 5, 5, 7, 3, 3, 7, 3, 6, 8, 5, 4, 8, 4, 5, 8, 4, 4, 5, 3, 5, 8, 6, 3, 6, 4, 5, 12, 5, 5, 5, 3, 6, 8, 5, 4, 8, 4, 4, 8, 4, 6, 9, 5
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OFFSET

1,4


COMMENTS

Conjecture: a(n)>0 for all n=3,4,...
The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
ZhiWei Sun also made the following similar conjectures:
(1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
(2) Each integer n>10 can be written as x+y (x,y>0) with 6x1 and 6x+1 both prime, and y and y+6 both practical.
Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x1 and 6x+1 both prime, and y and y+2 both practical.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 20122017.


EXAMPLE

a(14)=2 since 2*141=27=11+16=23+4, where 11 and 23 are primes, 16,16+2,4,4+2 are practical numbers.


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<3Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n12k]==True, 1, 0], {k, 1, n1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A000040, A005153, A199920.
Sequence in context: A064131 A111497 A220554 * A209320 A097051 A323761
Adjacent sequences: A208240 A208241 A208242 * A208244 A208245 A208246


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 11 2013


STATUS

approved



