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

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, 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, 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

Offset: 1

Author

Labos Elemer, Apr 07 2003

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000203, A009194, A020639.

Cf. also A082062, A082067, A082069, A082070, A082071, A082072.

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_] := n; f2[x_] := DivisorSigma[1, x]; Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {#, DivisorSigma[1, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

PARI

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082068(n) = A020639(gcd(sigma(n), n)); \\ Antti Karttunen, Nov 03 2017

Formula

a(n) = A020639(A009194(n)). - Antti Karttunen, Nov 03 2017

Extensions

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