A262766 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.
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, 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, 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
Offset: 1
Keywords
Examples
a(4) = 1 since 4 = 1^2 + 1^2 + phi(2^2) with 2*1*1 even and phi(1^2) < 4. 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. 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. 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.
Programs
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Mathematica
f[n_]:=EulerPhi[n^2] SQ[n_]:=IntegerQ[Sqrt[n]] 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}]
Comments