

A209315


Number of ways to write 2n1 = p+q with q practical, p and qp 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
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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+(qp), the conjecture implies Lemoine's conjecture related to A046927.
ZhiWei 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

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(9)=1 since 2*91=5+12 with 12 practical, 5 and 125 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<3Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[p]==True&&pr[2n1p]==True&&PrimeQ[2n12p]==True, 1, 0], {p, 1, n1}]
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

ZhiWei Sun, Jan 19 2013


STATUS

approved



