A233296 a(n) = |{0 < k < n: k*prime(n-k) + 1 is prime}|.

0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 4, 2, 2, 5, 4, 1, 6, 1, 2, 5, 4, 4, 4, 3, 3, 2, 4, 6, 5, 4, 5, 6, 7, 7, 6, 5, 8, 6, 4, 5, 7, 8, 4, 6, 7, 6, 10, 7, 4, 8, 11, 9, 11, 6, 5, 5, 8, 8, 10, 5, 7, 10, 10, 11, 7, 6, 6, 12, 6, 10, 11, 6, 11, 7, 11, 8, 12, 7, 7, 9, 13, 9, 10, 14, 7, 13, 8, 10, 11, 9, 14, 10, 14, 17, 14, 13, 8, 12, 12

Offset: 1

Author

Zhi-Wei Sun, Dec 07 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Similarly, for any integer n > 2, k*prime(n-k) - 1 (or k^2*prime(n-k) - 1) is prime for some 0 < k < n.

(ii) Let n > 3 be an integer. Then k + prime(n-k) is prime for some 0 < k < n. Also, if n is not equal to 13, then k^2 + prime(n-k)^2 is prime for some 0 < k < n.

Examples

			a(17) = 1 since 17 = 14 + 3 with 14*prime(3) + 1 = 14*5 + 1 = 71 prime.
a(19) = 1 since 19 = 18 + 1 with 18*prime(1) + 1 = 18*2 + 1 = 37 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Crossrefs

Cf. A000040, A232465, A232502, A233150, A233183, A233204, A233206.

Programs

Mathematica

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