A262746 Number of ordered ways to write n as x^2 + y^2 + pi(z^2) with 0 <= x <= y and z > 0 such that 2*x*y + 3 is prime, where pi(m) denotes the number of primes not exceeding m.

1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 5, 1, 4, 4, 3, 3, 6, 5, 2, 4, 4, 6, 3, 2, 5, 6, 3, 1, 6, 4, 4, 4, 4, 4, 4, 4, 2, 6, 4, 3, 7, 5, 5, 4, 6, 5, 7, 2, 3, 8, 3, 5, 3, 4, 6, 7, 5, 4, 7, 4, 6, 7, 3, 4, 8, 7, 4, 3, 4, 4, 11, 3, 4, 9, 4, 4, 6, 7, 2, 9, 6, 3, 6, 4, 6, 7, 3, 5, 8, 5, 5

Offset: 1

Author

Zhi-Wei Sun, Sep 29 2015

Keywords

• nonn

Comments

Conjectures:

(i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 21, 37, 117, 184. Also, any integer n > 8 can be written as x^2 + y^2 + pi(z^2), where x, y and z are integers with x+y (or z) odd.

(ii) Each n = 8,9,... can be written as p^2 + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.

(iii) Every n = 8,9,... can be written as pi(p^2) + pi(x^2) + pi(y^2), where p is prime, and x and y are positive integers.

Note that pi(x^2) > n if x > n > 0. We have verified that a(n) > 0 for all n = 1,...,10^6.

Examples

			a(1) = 1 since 1 = 0^2 + 1^2 + pi(1^2) with 2*0*1 + 3 = 3 prime.
a(2) = 2 since 2 = 0^2 + 0^2 + pi(2^2) = 1^2 + 1^2 + pi(1^2) with 2*0*0 + 3 = 3 and 2*1*1 + 3 = 5 both prime.
a(3) = 1 since 3 = 0^2 + 1^2 + pi(2^2) with 2*0*1 + 3 = 3 prime.
a(21) = 1 since 21 = 1^2 + 4^2 + pi(3^2) with 2*1*4 + 3 = 11 prime.
a(37) = 1 since 37 = 1^2 + 5^2 + pi(6^2) with 2*1*5 + 3 = 13 prime.
a(117) = 1 since 117 = 0^2 + 5^2 + pi(22^2) with 2*0*5 + 3 = 3 prime.
a(184) = 1 since 184 = 0^2 + 13^2 + pi(7^2) with 2*0*13 + 3 = 3 prime.
		

References

• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

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 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A000720, A001481, A038107, A262408, A262447, A262698, A262731.

Programs

Mathematica

pi[n_]:=PrimePi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[pi[z]>n,Goto[aa]];Do[If[SQ[n-pi[z]-y^2]&&PrimeQ[2y*Sqrt[n-pi[z]-y^2]+3],r=r+1],{y,0,Sqrt[(n-pi[z])/2]}];Continue,{z,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]