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A219026 Number of primes p<=n such that 2n-p and 2n+p-2 are both prime 3
0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 3, 3, 3, 3, 3, 1, 3, 1, 3, 2, 2, 2, 3, 4, 1, 4, 2, 2, 5, 1, 2, 4, 0, 3, 2, 3, 3, 2, 2, 1, 3, 3, 4, 4, 2, 3, 5, 3, 4, 3, 2, 3, 4, 3, 1, 3, 4, 3, 4, 3, 5, 4, 3, 3, 2, 3, 3, 3, 4, 2, 6, 3, 2, 7, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Conjecture: a(n)>0 except for n=1,2,4,6,10,22,57.
This is stronger than the Goldbach conjecture; it has been verified for n up to 5*10^7.
Zhi-Wei Sun also conjectured that if n is not among 1,2,3,5,8,87,108 then there is a prime p in (n,2n)
such that 2n-p and 2n+p-2 are both prime. For conjectures in Section 2 of arXiv:1211.1588, he had similar conjectures with p<=n replaced by p in (n,2n)
For example, if n is not among 1,2,4,6,10,15 then there is a prime p in (n,2n) such that
2n-p and 2n+p+2 are both prime.
LINKS
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588v2.
EXAMPLE
a(8)=2 since 3 and 5 are the only primes p<=8 with 16-p and 14+p both prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[2n-Prime[k]]==True&&PrimeQ[2n+Prime[k]-2]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 20000}]
np[n_]:=Count[Prime[Range[PrimePi[n]]], _?(AllTrue[{2n-#, 2n+#-2}, PrimeQ]&)]; Array[np, 100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 23 2017 *)
CROSSREFS
Sequence in context: A359290 A092928 A321090 * A085097 A374132 A117997
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 10 2012
STATUS
approved

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Last modified July 19 12:08 EDT 2024. Contains 374394 sequences. (Running on oeis4.)