A082054 Sum of common prime divisors (without multiplicity) of sigma(n) and phi(n).

0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 5, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0, 5, 2, 2, 2, 2, 2, 0, 2, 2, 5, 0, 2, 5, 2, 2, 2, 5, 2, 2, 5, 2, 2, 2, 3, 0, 2, 2, 2, 5, 2, 5, 2, 2, 2, 2, 2, 5, 2, 0, 5, 2, 2, 2, 2, 5, 2, 3, 2, 5, 2, 2, 5, 5, 2, 2, 0, 2, 2, 2, 2, 5, 2, 7, 2, 5, 2, 2, 2, 2, 5, 2, 2, 3, 5, 0, 2, 2, 2, 5, 5

Offset: 1

Author

Labos Elemer, Apr 03 2003

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000203, A008472, A009223.

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[Apply[Plus, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}]
a[n_] := Module[{g = GCD[DivisorSigma[1, n], EulerPhi[n]]}, If[g == 1, 0, Total[FactorInteger[g][[;; , 1]]]]]; Array[a, 100] (* Amiram Eldar, Feb 16 2025 *)

PARI

a(n)=my(f=factor(gcd(sigma(n=factor(n)), eulerphi(n)))[,1]); sum(i=1,#f,f[i]) \\ Charles R Greathouse IV, Dec 09 2013

Formula

a(n) = A008472(A009223(n)). - Amiram Eldar, Feb 16 2025