A219023 Number of primes p

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2, 1, 1, 0, 2, 1, 0, 2, 3, 0, 2, 2, 0, 1, 4, 1, 2, 1, 0, 0, 3, 1, 1, 3, 0, 0, 1, 2, 1, 1, 1, 1, 0, 0, 2, 3, 1, 0, 3, 1, 2, 1, 0, 1, 4, 0, 1, 2, 0, 2, 3, 0, 0, 4, 0, 2, 2, 0, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 3, 1, 2, 1, 0, 2, 4, 2

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>2732.

We have verified this conjecture for n up to 1.4*10^7. Note that the conjecture is stronger than Oppermann's conjecture which states that for any integer n>1 both of the two intervals (n^2-n,n^2) and (n^2,n^2+n) contain primes.

Zhi-Wei Sun also made the following conjectures: For n>3512 there is a prime p in (n,2n) such that both n^2-n+p and n^2+n-p are prime. For n>1828 there is a prime p4517 there is a prime in (n,2n) such that both n^2-n-p and n^2+n+p are prime.

Examples

			a(12)=2 since the 5 and 7 are the only primes p<12 with 12^2-12+p and 12^2+12-p both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
• Wikipedia, Oppermann's conjecture

Crossrefs

Cf. A000040.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[n^2-n+Prime[k]]==True&&PrimeQ[n^2+n-Prime[k]]==True,1,0],{k,1,PrimePi[n-1]}]
Do[Print[n," ",a[n]],{n,1,20000}]
Table[Total[Table[If[AllTrue[{k^2-k+p,k^2+k-p},PrimeQ],1,0],{p,Prime[ Range[ PrimePi[k]]]}]],{k,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 23 2017 *)

PARI

A219023(n)={my(c=0,nm=n^2-n,np=n^2+n); forprime(p=1,n-1,isprime(np-p) && isprime(nm+p) && c++); c} \\ - M. F. Hasler, Nov 11 2012