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Number of primes p<n such that n^2-n+p and n^2+n-p are both prime.
5

%I #20 Sep 23 2023 03:40:01

%S 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,

%T 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,

%U 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

%N Number of primes p<n such that n^2-n+p and n^2+n-p are both prime.

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

%C 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.

%C 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 p<n such that both n^2-n-p and n^2+n+p are prime. For n>4517 there is a prime in (n,2n) such that both n^2-n-p and n^2+n+p are prime.

%H Zhi-Wei Sun, <a href="/A219023/b219023.txt">Table of n, a(n) for n = 1..20000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann&#39;s_conjecture">Oppermann's conjecture</a>

%e 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.

%t 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]}]

%t Do[Print[n," ",a[n]],{n,1,20000}]

%t 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 *)

%o (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

%Y Cf. A000040.

%K nonn

%O 1,12

%A _Zhi-Wei Sun_, Nov 10 2012