cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Apr 07 2003

Keywords

Links

Crossrefs

Cf. A001157, A020639, A179930.
Cf. also A082063, A082067, A082068, 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[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 *)
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082069(n) = A020639(gcd(sigma(n,2), n)); \\ Antti Karttunen, Nov 03 2017

Formula

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

Extensions

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

A082067 Smallest prime that divides n and phi(n)=A000010(n), or 1 if n and phi(n) are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 5, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 7, 2, 1, 2, 1, 2, 5, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3
Offset: 1

Views

Author

Labos Elemer, Apr 07 2003

Keywords

Links

Crossrefs

Cf. A000010, A009195.
Cf. also A082061, A082068, 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_] := EulerPhi[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}]]] & @@ {#, EulerPhi@ #} &, 105] (* Michael De Vlieger, Nov 03 2017 *)
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082067(n) = A020639(gcd(eulerphi(n), n)); \\ Antti Karttunen, Nov 03 2017

Formula

a(n) = A020639(A009195(n)). - Antti Karttunen, Nov 03 2017

Extensions

Name clarified by Antti Karttunen, Nov 03 2017

A082070 Smallest prime that divides phi(n) and sigma(n) = A000203(n), or 1 if phi(n) and sigma(n) are relatively prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer, Apr 07 2003

Keywords

Links

Crossrefs

Cf. A000010, A000203, A009223, A020639.
Cf. also A082064, A082067, A082068, A082069, 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_] := EulerPhi[x]; 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}]]] & @@ {EulerPhi@ #, DivisorSigma[1, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082070(n) = A020639(gcd(eulerphi(n),sigma(n))); \\ Antti Karttunen, Nov 03 2017

Formula

a(n) = A020639(A009223(n)). - Antti Karttunen, Nov 03 2017

Extensions

Name edited by Antti Karttunen after an example by N. J. A. Sloane, Nov 04 2017

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Apr 07 2003

Keywords

Links

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

A082071 Smallest common prime-divisor of phi(n) = A000010(n) and sigma_2(n) = A001157(n); a(n)=1 if no common prime-divisor exists.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Labos Elemer, Apr 07 2003

Keywords

Links

Crossrefs

Cf. A000010, A001157, A020639.
Cf. also A082065, A082067, A082068, A082069, A082070, A082072.

Programs

  • Mathematica
    Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {EulerPhi@ #,
DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082071(n) = A020639(gcd(eulerphi(n),sigma(n,2))); \\ Antti Karttunen, Nov 03 2017

Formula

a(n) = A020639(gcd(A000010(n), A001157(n))). - Antti Karttunen, Nov 03 2017

Extensions

Values corrected by R. J. Mathar, Jul 09 2011
More terms from Antti Karttunen, Nov 03 2017
Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.