A238585 Number of primes p < n with prime(p)^2 + (prime(n)-1)^2 prime.

0, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 1, 3, 2, 3, 2, 2, 3, 1, 5, 1, 1, 3, 2, 4, 5, 2, 4, 3, 4, 1, 4, 5, 3, 4, 6, 3, 2, 2, 2, 2, 1, 8, 1, 3, 4, 7, 2, 5, 3, 2, 2, 4, 7, 4, 3, 2, 3, 5, 7, 5, 3, 6, 6, 5, 3, 4, 5, 2, 2, 2, 3, 7, 2, 3, 7, 3, 4, 10, 3

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 unless n divides 6, and a(n) = 1 only for n = 4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44.

(ii) If n > 2 is not equal to 9, then prime(n)^2 + (prime(p) - 1)^2 is prime for some prime p < n.

(iii) For n > 3, there is a prime p < n with prime(p) + prime(n) + 1 prime. If n > 9 is not equal to 18, then prime(p)^2 + prime(n)^2 - 1 is prime for some prime p < n.

Examples

			a(7) = 1 since 3 and prime(3)^2 + (prime(7)-1)^2 = 5^2 + 16^2 = 281 are both prime.
a(44) = 1 since 23 and prime(23)^2 + (prime(44)-1)^2 = 83^2 + 192^2 = 43753 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A232465, A238580.

Programs

Mathematica

p[n_,k_]:=PrimeQ[k]&&PrimeQ[Prime[k]^2+(Prime[n]-1)^2]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]