A237203 Least positive integer k < n/2 with phi(k)*phi(n-k) a square, or 0 if such a number k does not exist.

0, 0, 1, 0, 0, 1, 2, 0, 1, 2, 1, 2, 1, 2, 5, 7, 5, 1, 2, 4, 6, 3, 3, 4, 6, 6, 3, 4, 12, 3, 4, 14, 1, 2, 1, 2, 5, 1, 2, 8, 1, 2, 16, 6, 5, 7, 10, 8, 1, 2, 17, 7, 5, 3, 4, 8, 3, 1, 2, 6, 1, 2, 7, 1, 2, 11, 3, 4, 12, 6

Offset

1,7

Comments

Conjecture: a(n) < sqrt(n)*log(2*n) for all n > 0.

We have verified this for n up to 2*10^5. Note that a(211) = 85 > sqrt(211)*log(211) and a(373) = 117 > sqrt(373)*log(373).

According to the conjecture in A236998, a(n) should be positive for all n > 8.

Links

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

Example

 a(7) = 2 since phi(2)*phi(7-2) = 1*4 = 2^2 but phi(1)*phi(7-1) = 2 is not a square.

Mathematica

SQ[k_, m_]:=IntegerQ[Sqrt[EulerPhi[k]*EulerPhi[m]]]
Do[Do[If[SQ[k, n-k], Print[n, " ", k]; Goto[aa]], {k, 1, (n-1)/2}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000010, A000290, A236998.

Sequence in context: A336499 A093998 A247918 * A339444 A029389 A025835

Adjacent sequences: A237200 A237201 A237202 * A237204 A237205 A237206

Keyword

nonn

Author

Zhi-Wei Sun, Feb 05 2014

Status

approved