A262976 Number of ordered ways to write n as 2^x + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

1, 2, 2, 3, 4, 4, 4, 6, 4, 6, 6, 7, 6, 7, 5, 6, 10, 5, 9, 10, 7, 7, 9, 9, 4, 12, 10, 9, 8, 7, 10, 9, 10, 7, 15, 10, 6, 13, 10, 9, 10, 16, 10, 10, 9, 8, 15, 9, 8, 15, 12, 12, 7, 12, 11, 14, 12, 8, 16, 6, 10, 11, 14, 8, 11, 17, 10, 16, 9, 13, 16, 15, 8, 18, 13, 10, 14, 10, 12, 16, 12, 13, 18, 11, 9, 17, 17, 9, 15, 16, 15, 9, 12, 12, 17, 12, 9, 21, 10, 11

Offset: 1

Author

Zhi-Wei Sun, Oct 05 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0.

(ii) Each positive integer can be written as 2^x + pi(y^2) + pi(z^2) with x >= 0, y > 0 and z > 0.

Examples

			a(1) = 1 since 1 = 2^0 + 0^2 + pi(1^2).
a(2) = 2 since 2 = 2^0 + 1^2 + pi(1^2) = 2 + 0^2 + pi(1^2).
a(3) = 2 since 3 = 2^0 + 0^2 + pi(2^2) = 2 + 1^2 + pi(1^2).
		

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. A000079, A000290, A000720, A038107, A262746, A262887.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[2^y>n-f[x],Goto[bb]];If[SQ[n-f[x]-2^y],r=r+1],{y,0,Log[2,n-f[x]]}];Label[bb];Continue,{x,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]