A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | x*y*z, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

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, 4, 2, 4, 2, 2, 3, 1, 4, 2, 1, 2, 5, 4, 1, 2, 2, 6, 3, 2, 4, 4, 3, 3, 4, 3, 3, 5, 3, 3, 4, 2, 6, 7, 3, 4, 4, 5, 2, 2, 5, 6, 6, 1, 5, 4, 4, 4, 6, 6, 1, 4, 4, 2, 4, 3, 5, 6, 3, 4, 5, 5, 4, 2, 2, 6, 5, 4, 6, 3, 3, 1, 5, 3, 2, 3

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 4, 5, 8, 13, 16, 23, 32, 35, 39, 68, 75, 96, 215, 219, 243, 363, 471, 723, 759, 923, 1443, 1551, 1839, 2739, 2883.

It is easy to see that all the numbers phi(n^2) = n*phi(n) (n = 1,2,3,...) are pairwise distinct. We have verified that a(n) > 0 for all n = 1,...,10^7.

See also A262311 for a related conjecture.

Links

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

Example

a(5) = 1 since 5 = 0^2 + 2^2 + phi(1^2).
a(8) = 1 since 8 = 0^2 + 0^2 + phi(4^2) with 0*0*4 even, and phi(1^2) = 1, phi(2^2) = 2, phi(3^2) = 6 all smaller than 8.
a(13) = 1 since 13 = 1^2 + 2^2 + phi(4^2).
a(32) = 1 since 32 = 2^2 + 4^2 + phi(6^2).
a(68) = 1 since 68 = 0^2 + 6^2 + phi(8^2).
a(96) = 1 since 96 = 0^2 + 8^2 + phi(8^2).
a(363) = 1 since 363 = 0^2 + 19^2 + phi(2^2).
a(471) = 1 since 471 = 0^2 + 19^2 + phi(11^2).
a(723) = 1 since 723 = 17^2 + 18^2 + phi(11^2).
a(759) = 1 since 759 = 9^2 + 26^2 + phi(2^2).
a(923) = 1 since 923 = 16^2 + 25^2 + phi(7^2).
a(1443) = 1 since 1443 = 19^2 + 24^2 +phi(23^2).
a(1551) = 1 since 1551 = 18^2 + 35^2 + phi(2^2).
a(1839) = 1 since 1839 = 3^2 + 30^2 + phi(31^2).
a(2739) = 1 since 2739 = 1^2 + 24^2 + phi(47^2).
a(2883) = 1 since 2883 = 21^2 + 44^2 + phi(23^2).

Mathematica

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

Crossrefs

Cf. A000010, A001481, A002618, A262311, A262746, A262750, A262766.

Sequence in context: A101221 A086291 A262766 * A016442 A076360 A246702

Adjacent sequences: A262744 A262745 A262746 * A262748 A262749 A262750

Keyword

nonn

Author

Zhi-Wei Sun, Sep 30 2015

Status

approved