A262766 - OEIS

%I #10 Oct 01 2015 06:33:01

%S 1,2,1,1,1,3,2,1,2,4,2,2,1,2,2,1,2,3,2,3,4,3,1,2,3,3,1,4,2,2,2,1,3,2,

%T 1,2,4,4,1,2,2,6,3,2,3,4,3,3,4,3,3,4,3,3,4,2,5,6,3,4,4,4,2,2,4,5,4,1,

%U 5,4,3,4,5,6,1,4,3,2,3,3,5,5,3,4,4,4,3,2,2,5,4,4,5,3,3,1,3,3,2,3

%N Number of positive integers z such that n - phi(z^2) = x^2 + y^2 for some integers x and y with x*y*z even and phi(k^2) < n for all 0 < k < z.

%C The conjecture in A262747 implies that a(n) > 0 for all n > 0.

%e a(4) = 1 since 4 = 1^2 + 1^2 + phi(2^2) with 2*1*1 even and phi(1^2) < 4.

%e a(9) = 2 since 9 - phi(1^2) = 2^2 + 2^2 with 2*2*1 even, and 9 - phi(4^2) = 0^2 + 1^2 with 0*1*4 even and phi(k^2) < 9 for all k = 1..3.

%e a(35) = 1 since 35 - phi(3^2) = 2^2 + 5^2 with 2*5*3 even and phi(1^2) < phi(2^2) < 35.

%e a(96) = 1 since 96 - phi(8^2) = 0^2 + 8^2 with 0*8*8 even and phi(k^2) < 96 for all k = 1..7.

%t f[n_]:=EulerPhi[n^2]

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[(Mod[x*y,2]==0||Mod[Sqrt[n-f[x]-y^2],2]==0)&&SQ[n-f[x]-y^2],r=r+1;Goto[bb]],{y,0,Sqrt[(n-f[x])/2]}];Label[bb];Continue,{x,1,n}]; Label[aa];Print[n," ",r];Continue,{n,1,100}]

%Y Cf. A000010, A001481, A002618, A262747, A262750.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Sep 30 2015