A082066 -id:A082066 - cp's OEIS frontend

A082061 Greatest common prime divisor of n and phi(n)=A000010(n); a(n)=1 if no common prime divisor exists.

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

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A000010, A003277, A006530, A009195, A070003.

Cf. also A082062, A082063, A082064, A082065, A082066, A082067.

gcpd := proc(a,b) local g ,d ; g := 1 ; for d in numtheory[divisors](a) intersect numtheory[divisors](b) do if isprime(d) then g := max(g,d) ; end if; end do: g ; end proc:
A082061 := proc(n) gcpd( numtheory[phi](n), n) ; end proc: # R. J. Mathar, Jul 09 2011

(* 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_] := EulerPhi[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}]]] & @@ {#, EulerPhi@ #} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

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

a(n) = A006530(A009195(n)). - Antti Karttunen, Nov 03 2017
From Amiram Eldar, Dec 06 2024: (Start)
a(n) <= A006530(n), with equality if and only if n is in A070003.
a(n) = 1 if and only if n is a cyclic number (A003277). (End)

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

A082065 Greatest 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, 5, 2, 2, 1, 2, 2, 3, 2, 2, 2, 1, 5, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 1, 2, 5, 2, 2, 2, 2, 2, 1, 2, 2, 5, 3, 5, 2, 2, 2, 1, 5, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 1

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006530, A001157, A000010, A082061-A082066.

gcpd := proc(a,b) local g ,d ; g := 1 ; for d in numtheory[divisors](a) intersect numtheory[divisors](b) do if isprime(d) then g := max(g,d) ; end if; end do: g ; end proc:
A082065 := proc(n) gcpd( numtheory[phi](n), numtheory[sigma][2](n) ) ; end proc:
seq(A082065(n),n=1..120) ; # R. J. Mathar, Jul 09 2011

Table[FactorInteger[GCD[EulerPhi@ n, DivisorSigma[2, n]]][[-1, 1]], {n, 100}] (* Michael De Vlieger, Jul 22 2017 *)

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
a(n)=gpf(gcd(eulerphi(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 21 2013

Values corrected by R. J. Mathar, Jul 09 2011

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

A082072 Smallest prime that divides sigma(n) = A000203(n) and sigma_2(n) = A001157(n), or 1 if sigma(n) and sigma_2(n) are relatively prime.

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 2, 5, 13, 2, 2, 2, 2, 2, 2, 31, 2, 13, 2, 2, 2, 2, 2, 2, 31, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 127, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 7, 2, 2

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A000203, A001157, A020639, A179931.

Cf. also A082066, A082067, A082068, A082069, A082070, A082071.

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, x]; 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[1, #], DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

lpf(n)=my(f=factor(n)[,1]); if(#f,f[1],1)
a(n)=lpf(gcd(sigma(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A020639(A179931(n)). - Antti Karttunen, Nov 03 2017

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

A082064 Greatest common prime-divisor of phi(n) and sigma(n) = A000203(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, 3, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 3, 3

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A000010, A000203, A006530, A009223.

Cf. also A082061, A082062, A082063, A082065, A082066, A082070.

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[n]; f2[x_] := DivisorSigma[1, 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}]]] & @@ {EulerPhi@ #, DivisorSigma[1, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A082064(n) = A006530(gcd(eulerphi(n), sigma(n))); \\ Antti Karttunen, Nov 03 2017

a(n) = A006530(A009223(n)). - Antti Karttunen, Nov 03 2017

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

cp's OEIS Frontend

A082061 Greatest common prime divisor of n and phi(n)=A000010(n); a(n)=1 if no common prime divisor exists.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A082065 Greatest common prime-divisor of phi(n)=A000010(n) and sigma(2,n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A082072 Smallest prime that divides sigma(n) = A000203(n) and sigma_2(n) = A001157(n), or 1 if sigma(n) and sigma_2(n) are relatively prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A082064 Greatest common prime-divisor of phi(n) and sigma(n) = A000203(n); a(n)=1 if no common prime-divisor exists.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions