OFFSET
1,5
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.
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.
EXAMPLE
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).
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]<f[k]||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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 20 2019
STATUS
approved