A238576 Number of odd primes p < 2*n with prime(n*(p-1)/2)^2 - 2 prime.

0, 1, 2, 3, 2, 3, 1, 4, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 4, 5, 2, 2, 1, 8, 2, 2, 3, 3, 2, 2, 4, 4, 5, 6, 2, 5, 4, 3, 3, 7, 2, 2, 8, 8, 5, 4, 6, 3, 3, 7, 6, 5, 3, 3, 9, 4, 8, 3, 5, 3, 1, 5, 6, 4, 6, 7, 7, 8, 6, 6, 2, 7, 1, 5, 9, 7, 5, 6, 5, 7

Offset: 1

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 7, 23, 61, 73.

(ii) For any integer n > 1, there is an odd prime p < 2*n with prime(n*(p+1)/2)^2 - 2 prime.

Clearly, either part of the conjecture implies that there are infinitely many primes of the form p^2 - 2 with p prime.

Examples

			a(2) = 1 since 2 and prime(2*(3-1)/2)^2 - 2 = 3^2 - 2 = 7 are both prime.
a(7) = 1 since 5 and prime(7*(5-1)/2)^2 - 2 = 43^2 - 2 = 1847 are both prime.
a(23) = 1 since 29 and prime(23*(29-1)/2)^2 - 2 = 2137^2 - 2 = 4566767 are both prime.
a(61) = 1 since 43 and prime(61*(43-1)/2)^2 - 2 = 10463^2 - 2 = 109474367 are both prime.
a(73) = 1 since 7 and prime(73*(7-1)/2)^2 - 2 = 1367^2 - 2 = 1868687 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, A049002, A062326, A237413, A237578, A238573.

Programs

Mathematica

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