A262750 Least positive integer 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, or 0 if no such z exists, where phi(.) is Euler's totient function given by A000010.

1, 1, 2, 2, 1, 1, 2, 4, 1, 1, 2, 2, 4, 1, 2, 4, 1, 1, 2, 2, 1, 2, 3, 4, 4, 1, 2, 2, 5, 1, 2, 6, 1, 2, 3, 2, 1, 1, 2, 4, 1, 1, 2, 4, 4, 1, 2, 4, 4, 1, 2, 2, 1, 1, 2, 5, 4, 3, 3, 2, 4, 1, 2, 6, 1, 1, 2, 8, 1, 2, 3, 4, 1, 1, 2, 2, 6, 3, 3, 4, 1, 1, 2, 2, 5, 1, 2, 4, 4, 1, 2, 2, 4, 6, 3, 8, 4, 1, 2, 2

Offset

1,3

Comments

Conjecture: a(n) <= sqrt(n) except for n = 3, 8, 13, 32.

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

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(68) = 8 since 68 - phi(8^2) = 68 - 32 = 36 = 0^2 + 6^2 with 0*6*8 even and all those phi(k^2) (k = 1,...,7) smaller than 68.
a(5403) = 67 since 5403 - phi(67^2) = 5403 - 4422 = 981 = 9^2 + 30^2 with 9*30*67 even and all those phi(k^2) (k = 1,...,5403) smaller than 5403.

Mathematica

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

Crossrefs

Cf. A000010, A001481, A002618, A233867, A262747.

Sequence in context: A331910 A240168 A199204 * A075402 A276696 A220777

Adjacent sequences: A262747 A262748 A262749 * A262751 A262752 A262753

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2015

Status

approved