A081396 Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).

0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2

Offset: 1

Author

Labos Elemer, Mar 28 2003

Keywords

• nonn

Examples

			n=209: sigma(209) = 240 = 2*2*2*2*3*5, phi(209) = 180 = 2*2*3*3*5, common factor set = {2,3,5}, so a(209)=3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A000203, A001221, A009223, A081383, A082055.

Programs

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Length[Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 100}]

PARI

a(n)=omega(gcd(sigma(n),eulerphi(n))) \\ Charles R Greathouse IV, Feb 19 2013

Formula

a(n) = A001221(A009223(n)). - Antti Karttunen, Jan 22 2020

Extensions

Data section extended up to a(105) by Antti Karttunen, Jan 22 2020