A081396 - OEIS

%I #16 Jan 22 2020 20:08:45

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A081396/b081396.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = A001221(A009223(n)). - _Antti Karttunen_, Jan 22 2020

%e 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.

%t 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}]

%o (PARI) a(n)=omega(gcd(sigma(n),eulerphi(n))) \\ _Charles R Greathouse IV_, Feb 19 2013

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

%K nonn

%O 1,14

%A _Labos Elemer_, Mar 28 2003

%E Data section extended up to a(105) by _Antti Karttunen_, Jan 22 2020