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A208243 Number of ways to write 2n-1 = 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 (list; graph; refs; listen; history; text; internal format)
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.

Zhi-Wei 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 6x-1 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 6x-1 and 6x+1 both prime, and y and y+2 both practical.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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(14)=2 since 2*14-1=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<3||Mod[n, 2]+Con[n]==0)

a[n_]:=a[n]=Sum[If[pr[2k]==True&&pr[2k+2]==True&&PrimeQ[2n-1-2k]==True, 1, 0], {k, 1, n-1}]

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

Zhi-Wei Sun, Jan 11 2013

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)