A232465 a(n) = |{0 < k <= n/2: prime(k) + prime(n-k) - 1 is prime}|.

0, 1, 0, 1, 1, 0, 2, 1, 3, 1, 3, 1, 3, 1, 3, 4, 5, 2, 5, 2, 5, 5, 4, 5, 4, 5, 6, 8, 2, 8, 9, 11, 4, 6, 1, 3, 6, 8, 8, 7, 3, 11, 9, 8, 8, 9, 12, 8, 10, 10, 10, 8, 6, 3, 8, 11, 13, 14, 13, 15, 8, 15, 15, 14, 8, 18, 11, 14, 5, 10, 7, 10, 15, 12, 10, 5, 10, 11, 12, 16, 21, 15, 16, 14, 8, 15, 19, 14, 16, 18, 13, 10, 28, 21, 14, 20, 18, 24, 20, 19

Offset: 1

Author

Zhi-Wei Sun, Nov 24 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 except for n = 1, 3, 6. Also, a(n) = 1 only for n = 2, 4, 5, 8, 10, 12, 14, 35.

(ii) For each integer n > 7, there is a positive integer k < n/2 with (prime(n-k) - prime(k))/2 prime. Also, for any positive integer n not among 1, 3, 5, 9, 21, (prime(k) + prime(n-k))/2 is prime for some 0 < k < n.

(iii) For any integer n > 6, prime(k)^2 + prime(n-k)^2 - 1 is prime for some 0 < k < n. Also, for any integer n > 4 not equal to 14, (prime(k)^2 + prime(n-k)^2)/2 is prime for some 0 < k < n.

(iv) For any integer n > 3, (prime(k) - 1)^2 + prime(n-k)^2 is prime for some 0 < k < n. Also, if n > 4 then (prime(k) + 1)^2 + prime(n-k)^2 is prime for some 0 < k < n.

Examples

			a(8) = 1 since prime(4) + prime(4) - 1 = 13 is prime.
a(10) = 1 since prime(4) + prime(6) - 1 = 7 + 13 - 1 = 19 is prime.
a(14) = 1 since prime(6) + prime(8) - 1 = 13 + 19 - 1 = 31 is prime.
a(35) = 1 since prime(2) + prime(33) - 1 = 3 + 137 - 1 = 139 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A232443, A232463.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[k]+Prime[n-k]-1],1,0],{k,1,n/2}]
Table[a[n],{n,1,100}]