A237706 Number of primes p < n with pi(n-p) a square, where pi(.) is given by A000720.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 11 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 5, 6, 7, 8, 16, 17, 22, 23, 148, 149.

(ii) For any integer n > 2, there is a prime p < n with pi(n-p) a triangular number.

We have verified that a(n) > 0 for every n = 3, ..., 1.5*10^7. See A237710 for the least prime p < n with pi(n-p) a square.

See also A237705, A237720 and A237721 for similar conjectures.

Examples

			a(8) = 1 since 7 is prime with pi(8-7) = 0^2.
a(16) = 1 since 7 is prime with pi(16-7) = 2^2.
a(149) = 1 since 139 is prime with pi(149-139) = pi(10) = 2^2.
a(637) = 2 since 409 is prime with pi(637-409) = pi(228) = 7^2, and 613 is prime with pi(637-613) = pi(24) = 3^2.
		

Links

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

Crossrefs

Cf. A000040, A000217, A000290, A000720, A237598, A237612, A237705, A237710, A237720, A237721.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
q[n_]:=SQ[PrimePi[n]]
a[n_]:=Sum[If[q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,70}]