A308342 Number of ways to write 2*n as phi(x^2) + phi(y^2) + phi(z^2), where x,y,z are positive integers with x <= y <= z, and phi(.) is Euler's totient function (A000010).

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

Offset: 1

Author

Zhi-Wei Sun, May 20 2019

Keywords

• nonn

Comments

Conjecture 1: a(n) > 0 for all n > 1. In other words, the set {phi(x^2) + phi(y^2) + phi(z^2): x,y,z = 1,2,3,...} contains all even numbers greater than two.

Conjecture 2: For any integer n > 3, we can write 2*n+1 as phi(x^2) + phi(y^2) + sigma(z^2) with x,y,z positive integers, where the function sigma(.) is given by A000203.

Examples

			a(2) = 1 with 2*2 = phi(1^2) + phi(1^2) + phi(2^2).
a(3) = 1 with 2*3 = phi(2^2) + phi(2^2) + phi(2^2).
a(4) = 1 with 2*4 = phi(1^2) + phi(1^2) + phi(3^2).
a(6) = 1 with 2*6 = phi(2^2) + phi(2^2) + phi(4^2).
a(19) = 1 with 2*19 = phi(3^2) + phi(5^2) + phi(6^2).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Does phi(x^2) + phi(y^2) + phi(z^2) represent all even numbers greater than two?, Question 332067 on MathOverflow, May 20, 2019.

Crossrefs

Cf. A000010, A000203, A000290, A002618, A262311, A262982, A262985.

Programs

Mathematica

f[n_]:=f[n]=n*EulerPhi[n]
T={};Do[If[f[n]<=200,T=Append[T,f[n]]],{n,1,200}];
tab={};Do[r=0;Do[If[f[k]>2n/3,Goto[cc]];Do[If[f[m](2n-f[k])/2,Goto[bb]];If[MemberQ[T,2n-f[k]-f[m]],r=r+1];Label[bb],{m,1,(2n-f[k])/2}];Label[cc],{k,1,2n/3}];tab=Append[tab,r],{n,1,100}];Print[tab]