cp's OEIS Frontend

A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime.

%I A220272 #19 Jul 23 2025 01:00:00
%S A220272 0,0,1,1,2,1,2,1,1,3,2,2,2,1,1,3,3,1,3,2,2,3,2,2,3,2,2,3,1,1,4,3,1,2,
%T A220272 2,2,5,3,3,3,4,3,3,1,3,3,2,2,4,4,2,6,2,2,4,4,2,3,1,2,5,4,1,3,3,3,6,2,
%U A220272 3,5,4,3,3,3,3,6,3,2,4,2,3,4,3,2,5,3,5,2,1,1,9,4,3,4,3,5,3,2,2,2
%N A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime.
%C A220272 Conjecture: a(n)>0 for all n>2.
%C A220272 This has been verified for n up to 3*10^9. The author observed that for each n=3,...,3*10^9 we may even require x<(log n)^2, but Jack Brennen found that for n=4630581798 we cannot require x<(log n)^2.
%C A220272 The author guessed that the conjecture can be slightly refined as follows: Any integer n>2 can be written as x^2+y with 2*x*y-1 prime, where x and y are positive integers with x<=y.
%C A220272 Zhi-Wei Sun also made the following general conjecture: If m is a positive integer and r is 1 or -1, then any sufficiently large integer n can be written as x^2+y (x>0, y>0) with m*x*y+r prime.
%C A220272 For example, for (m,r)=(1,-1),(1,1),(2,1),(3,-1),(3,1),(4,-1),(4,1),(5,-1),(5,1),(6,-1),(6,1), it suffices to require that n is greater than 12782, 15372, 488, 5948, 2558, 92, 822, 21702, 6164, 777, 952 respectively.
%H A220272 Zhi-Wei Sun, <a href="/A220272/b220272.txt">Table of n, a(n) for n = 1..20000</a>
%H A220272 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%H A220272 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b1fbebe8.1212">Re: A curious conjecture on primes</a>, a message to Number Theory List, Dec. 12, 2012.
%e A220272 a(18)=1 since 18=3^2+9 with 2*3*9-1=53 prime.
%t A220272 a[n_]:=a[n]=Sum[If[PrimeQ[2k(n-k^2)-1]==True,1,0],{k,1,Sqrt[n]}]
%t A220272 Do[Print[n," ",a[n]],{n,1,100}]
%Y A220272 Cf. A219842, A219864, A219923.
%K A220272 nonn
%O A220272 1,5
%A A220272 _Zhi-Wei Sun_, Dec 09 2012