A235703 Number of ordered ways to write n = p + q with p a term of A234695 and q a term of A235592.

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

Offset: 1

Author

Zhi-Wei Sun, Jan 14 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 30, 83.

Examples

			 a(4) = 1 since 4 = 2 + 2 with 2, prime(2) - 2 + 1 = 2 and 2*3 - prime(2) = 3 all prime.
a(30) = 1 since 30 = 3 + 27 with 3, prime(3) - 3 + 1 = 3 and 27*28 - prime(27) = 756 - 103 = 653 all prime.
a(83) = 1 since 83 = 13 + 70 with 13, prime(13) - 13 + 1 = 29 and 70*71 - prime(70) = 4970 - 349 = 4621 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A234695, A235330, A235508, A235592.

Programs

Mathematica

p[n_]:=PrimeQ[Prime[n]-n+1]
q[n_]:=PrimeQ[n(n+1)-Prime[n]]
a[n_]:=Sum[If[p[Prime[k]]&&q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]