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A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime. 9
0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3, 1, 3, 4, 2, 2, 2, 3, 4, 3, 1, 3, 3, 1, 4, 5, 3, 3, 3, 2, 5, 4, 1, 3, 5, 2, 5, 4, 3, 4, 5, 2, 5, 5, 2, 4, 5, 3, 6, 5, 5, 5, 2, 3, 6, 5, 2, 3, 4, 3, 6, 5, 4, 4, 4, 5, 6, 6, 4, 5, 4, 3, 6, 8, 2, 2, 5, 6, 7, 6, 2, 6, 2, 4, 7, 6, 4, 3, 6, 3, 5, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

Conjecture: a(n)>0 for all n>8.

This has been verified for n up to 10^7.

As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927.

Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime,  one of q and q+1 is prime and another of q and q+1 is 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(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime.

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[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True, 1, 0], {p, 1, n-1}]

Do[Print[n, " ", a[n]], {n, 1, 100}]

CROSSREFS

Cf. A005153, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185.

Sequence in context: A072504 A072499 A060272 * A174713 A129985 A085243

Adjacent sequences:  A209312 A209313 A209314 * A209316 A209317 A209318

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 19 2013

STATUS

approved

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Last modified October 20 02:52 EDT 2019. Contains 328244 sequences. (Running on oeis4.)