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

1,2

Comments

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

Links

Table of n, a(n) for n=1..100.

Example

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.

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}]

Crossrefs

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

Sequence in context: A344756 A101221 A086291 * A262747 A016442 A076360

Adjacent sequences: A262763 A262764 A262765 * A262767 A262768 A262769

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2015

Status

approved