A179930 -id:A179930 - cp's OEIS frontend

A082069 Smallest common prime-divisor of n and Sigma_2(n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 5, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 7, 1, 5, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 5

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A001157, A020639, A179930.

Cf. also A082063, A082067, A082068, A082070, A082071, A082072.

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[2, 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[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082069(n) = A020639(gcd(sigma(n,2), n)); \\ Antti Karttunen, Nov 03 2017

a(n) = A020639(A179930(n)). - Antti Karttunen, Nov 03 2017

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

A179931 a(n) = gcd(sigma(n), sigma_2(n)).

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 2, 5, 13, 2, 2, 14, 2, 2, 4, 31, 2, 13, 2, 42, 4, 2, 2, 10, 31, 2, 20, 14, 2, 4, 2, 21, 4, 2, 4, 91, 2, 10, 4, 10, 2, 4, 2, 42, 26, 2, 2, 62, 57, 93, 4, 14, 2, 20, 4, 10, 20, 10, 2, 84, 2, 2, 26, 127, 4, 4, 2, 42, 4, 4, 2, 65, 2, 2, 62, 14, 4, 4, 2, 62, 121, 2, 2, 28, 4, 2, 20, 10, 2, 26, 4, 42, 4, 2, 4, 42, 2, 57, 26, 217

Offset: 1

N. J. A. Sloane, Jul 09 2011, following a suggestion from R. J. Mathar

nonn

A006530(a(n)) = A082066(n). - Reinhard Zumkeller, Jul 10 2011, the latter A-number corrected by Antti Karttunen, May 22 2017.

Not multiplicative: a(2)*a(19) <> a(38). - R. J. Mathar, Oct 08 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A179930, A000203, A001157, A082066, A009194.

A179931 := proc(n) igcd( numtheory[sigma](n), numtheory[sigma][2](n)) ; end proc:
seq(A179931(n),n=1..100) ;

Table[GCD @@ Map[DivisorSigma[#, n] &, {1, 2}], {n, 100}] (* Michael De Vlieger, May 22 2017 *)

a(n)=gcd(sigma(n),sigma(n,2)) \\ Charles R Greathouse IV, Feb 14 2013

A082063 Greatest common prime divisor of n and sigma_2(n) = A001157(n), or 1 if the two are relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 7, 1, 5, 1, 1, 1, 2, 5, 3, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 7, 1, 13, 2, 1, 2, 1, 5, 1, 1, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 7, 5, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 5

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A006530, A001157, A179930.

Cf. also A082061, A082062, A082064, A082065, A082066, A082069.

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

A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A082063(n) = A006530(gcd(sigma(n,2), n)); \\ Antti Karttunen, Nov 03 2017

a(n) = A006530(A179930(n)). - Antti Karttunen, Nov 03 2017

Erroneous comment removed by Antti Karttunen, Nov 03 2017

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A082069 Smallest common prime-divisor of n and Sigma_2(n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

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A179931 a(n) = gcd(sigma(n), sigma_2(n)).

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A082063 Greatest common prime divisor of n and sigma_2(n) = A001157(n), or 1 if the two are relatively prime.

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