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A082068 Smallest common prime-divisor of n and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.

%I A082068 #14 Jan 29 2022 13:10:15
%S A082068 1,1,1,1,1,2,1,1,1,2,1,2,1,2,3,1,1,3,1,2,1,2,1,2,1,2,1,2,1,2,1,1,3,2,
%T A082068 1,1,1,2,1,2,1,2,1,2,3,2,1,2,1,1,3,2,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,2,
%U A082068 3,2,1,3,1,2,1,2,1,2,1,2,1,2,1,2,1,2,3,2,1,2,7,2,1,2,5,2,1,1,3,1,1,2,1,2,3
%N A082068 Smallest common prime-divisor of n and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.
%H A082068 Antti Karttunen, <a href="/A082068/b082068.txt">Table of n, a(n) for n = 1..16384</a>
%F A082068 a(n) = A020639(A009194(n)). - _Antti Karttunen_, Nov 03 2017
%t A082068 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_] := n; f2[x_] := DivisorSigma[1, x]; Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
%t A082068 (* Second program: *)
%t A082068 Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {#, DivisorSigma[1, #]} &, 105] (* _Michael De Vlieger_, Nov 03 2017 *)
%o A082068 (PARI)
%o A082068 A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
%o A082068 A082068(n) = A020639(gcd(sigma(n), n)); \\ _Antti Karttunen_, Nov 03 2017
%Y A082068 Cf. A000203, A009194, A020639.
%Y A082068 Cf. also A082062, A082067, A082069, A082070, A082071, A082072.
%K A082068 nonn
%O A082068 1,6
%A A082068 _Labos Elemer_, Apr 07 2003
%E A082068 Changed "was found" to "exists" in definition. - _N. J. A. Sloane_, Jan 29 2022