A082066 Greatest common prime-divisor of sigma_1(n)=A000203(n) and sigma_2(n)=A001157(n); a(n)=1 if no common prime-divisor exists.

1, 1, 2, 7, 2, 2, 2, 5, 13, 2, 2, 7, 2, 2, 2, 31, 2, 13, 2, 7, 2, 2, 2, 5, 31, 2, 5, 7, 2, 2, 2, 7, 2, 2, 2, 13, 2, 5, 2, 5, 2, 2, 2, 7, 13, 2, 2, 31, 19, 31, 2, 7, 2, 5, 2, 5, 5, 5, 2, 7, 2, 2, 13, 127, 2, 2, 2, 7, 2, 2, 2, 13, 2, 2, 31, 7, 2, 2, 2, 31, 11, 2, 2, 7, 2, 2, 5, 5, 2, 13, 2, 7, 2, 2, 2, 7, 2

Offset: 1

Author

Labos Elemer, Apr 07 2003

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006530, A001157, A000203, A082061-A082065.

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]}] f1[x_] := DivisorSigma[1, n]; f2[x_] := DivisorSigma[2, x] Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Table[Last[Apply[Intersection, FactorInteger[Map[DivisorSigma[#, n] &, {1, 2}]][[All, All, 1]]] /. {} -> {1}], {n, 109}] (* Michael De Vlieger, May 22 2017 *)

PARI

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
a(n)=gpf(gcd(sigma(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 19 2013

Python

from sympy import primefactors, gcd, divisor_sigma
def a006530(n): return 1 if n==1 else primefactors(n)[-1]
def a(n): return a006530(gcd(divisor_sigma(n), divisor_sigma(n, 2))) # Indranil Ghosh, May 22 2017

Formula

a(n) = A006530(A179931(n)). - Reinhard Zumkeller, Jul 10 2011

Extensions

Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022