A009223 -id:A009223 - cp's OEIS frontend

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 6, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 2, 8, 2, 2, 2, 2, 2, 8, 6, 2, 2, 8, 6, 4, 2, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 4, 24, 2, 2, 2, 6, 8, 6, 12, 24, 2, 2, 2, 2, 2, 12, 4, 6, 8, 2, 2, 8, 8, 2, 4, 2, 24, 2, 2, 6, 4

Offset: 1

Labos Elemer, Dec 04 2001

nonn

Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).

Antti Karttunen, Table of n, a(n) for n = 1..23374
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).

a(n) = A322360(n) / A322359(n). - Antti Karttunen, Dec 04 2018

Name edited, part of the old name transferred to the formula section by Antti Karttunen, Dec 04 2018

A055008 Numbers k such that gcd(phi(k), sigma(k)) = 1 with phi = A000010, sigma = A000203.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 36, 50, 64, 81, 100, 121, 128, 144, 225, 242, 256, 289, 324, 400, 484, 512, 529, 576, 578, 625, 729, 800, 841, 900, 1024, 1058, 1089, 1156, 1250, 1296, 1600, 1681, 1682, 1936, 2025, 2048, 2116, 2209, 2304, 2312, 2401, 2500, 2601

Offset: 1

Labos Elemer, May 31 2000

nonn

The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020
Conjecture: Every term is a square or twice a square. - Jason Yuen, May 16 2024
The conjecture is true: If k is neither a square nor twice a square (i.e., in A028983), then sigma(k) is even. Since gcd(phi(k), sigma(k)) = 1, then phi(k) must be odd, but phi(k) is odd only for k = 1 and 2. - Amiram Eldar, May 19 2024

			For n = 484, phi(484) = 220 = 2*2*5*11, sigma(484) = 931 = 7*7*19, and gcd(220,931) = 1.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.

Cf. A000010, A000203, A009223, A028983.

Select[Range@ 2700, CoprimeQ[EulerPhi@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Feb 05 2017 *)
Select[With[{max = 51}, Union[Array[#^2 &, max], Array[2*#^2 &, Floor[max / Sqrt[2]]]]], CoprimeQ[EulerPhi[#], DivisorSigma[1, #]] &] (* Amiram Eldar, May 19 2024 *)

is(n)=gcd(sigma(n),eulerphi(n))==1 \\ Charles R Greathouse IV, Feb 19 2013

Incorrect comment removed by Charles R Greathouse IV, Feb 19 2013

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

Labos Elemer, Apr 07 2003

nonn

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

Cf. A000010, A000203, A009223, A020639.

Cf. also A082064, A082067, A082068, A082069, 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_] := 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 *)

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082070(n) = A020639(gcd(eulerphi(n),sigma(n))); \\ Antti Karttunen, Nov 03 2017

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

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

A066087 a(n) = gcd(sigma(n), phi(n)) - gcd(sigma(rad(n)), phi(rad(n))); rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 1, -1, 0, -4, 0, 4, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -4, -2, 0, 0, 0, 0, -1, 0, 0, -4, 0, 0, 0, 18, 0, -2, 0, 2, 0, 0, 0, 2, 0, -3, 8, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 12, -6

Offset: 1

Labos Elemer, Dec 04 2001

sign

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

Cf. A048250, A023900, A000203, A007947, A000010, A009223, A066086.

Table[GCD[DivisorSigma[1, n], EulerPhi@ n] - GCD[DivisorSigma[1, #], EulerPhi@ #] &[Times @@ FactorInteger[n][[All, 1]]], {n, 120}] (* Michael De Vlieger, Feb 19 2017 *)

rad(f)=for(i=1,#f~,f[i,2]=1); f
g(f)=gcd(sigma(f),eulerphi(f))
a(n)=my(f=factor(n),k=rad(f)); g(f)-g(k) \\ Charles R Greathouse IV, Dec 09 2013

A009223(n) - A066086(n) = gcd(sigma(n), phi(n)) - gcd(sigma(A007947(n)), phi(A007947(n))).

A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

Original entry on oeis.org

1, 3, 18, 12, 200, 14, 3364, 15, 722, 328, 9801, 42, 25281, 116, 1800, 165, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 172, 98942809, 488, 1547536, 336, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 319, 3150464641

Offset: 1

Labos Elemer, Nov 12 2002

nonn

Values are frequently identical to terms of A077102. Since gcd(a,b) and gcd(a+b,a-b) may differ, so may the smallest solutions. A077102(m) and a(m) differ at m = 1, 2, 4, 8, 16, 28, 32, 40, etc.

Giovanni Resta, Table of n, a(n) for n = 1..120

Cf. A000203, A000010, A009223, A055008, A077099-A077102, A051612, A065387, A023897, A020492.

f[x_] := Apply[GCD, {DivisorSigma[1, x], EulerPhi[x]}] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}];

a(n)=my(x=n);while(gcd(eulerphi(x),sigma(x))!=n, x++); x \\ Charles R Greathouse IV, Dec 09 2013

a(n) = Min{x; A055008(x)=n}. a(n)=Min{x; gcd(A000203(x), A000010(x))=n}

a(n) = Min{x: A023897(x)= n}, smallest balanced number (A020492) for which the quotient equals n.

A074389 a(n) = gcd(n, sigma(n), phi(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 1, 2, 1, 6, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 4, 1, 1, 3, 1, 1, 2, 1, 2, 3

Offset: 1

Labos Elemer, Aug 23 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A000203, A073815, A050399, A009194, A009195, A009223, A074465, A074466.

In the old definition the erroneously given formula gcd(n, A000005(n), A000010(n)) is now sequence A318459. - Antti Karttunen, Sep 07 2018

Table[Apply[GCD, {w, DivisorSigma[1, w], EulerPhi[w]}], {w, 1, 128}]

A074389(n) = gcd([n, sigma(n), eulerphi(n)]); \\ Antti Karttunen, Sep 07 2018

a(n) = gcd(n, A000010(n), A000203(n)).

a(n) = gcd(n, A009223(n)). - Antti Karttunen, Sep 07 2018

Name corrected by Antti Karttunen, Sep 07 2018

A081396 Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2

Offset: 1

Labos Elemer, Mar 28 2003

nonn

			n=209: sigma(209) = 240 = 2*2*2*2*3*5, phi(209) = 180 = 2*2*3*3*5, common factor set = {2,3,5}, so a(209)=3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A000203, A001221, A009223, A081383, A082055.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Length[Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 100}]

a(n)=omega(gcd(sigma(n),eulerphi(n))) \\ Charles R Greathouse IV, Feb 19 2013

a(n) = A001221(A009223(n)). - Antti Karttunen, Jan 22 2020

Data section extended up to a(105) by Antti Karttunen, Jan 22 2020

A009286 a(n) = lcm(sigma(n), phi(n)).

Original entry on oeis.org

1, 3, 4, 14, 12, 12, 24, 60, 78, 36, 60, 28, 84, 24, 24, 248, 144, 78, 180, 168, 96, 180, 264, 120, 620, 84, 360, 168, 420, 72, 480, 1008, 240, 432, 48, 1092, 684, 180, 168, 720, 840, 96, 924, 420, 312, 792, 1104, 496, 798, 1860, 288, 1176, 1404, 360, 360, 120, 720, 1260

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A000203, A009223, A062354, A084921.

A009286(n) = lcm(sigma(n), eulerphi(n)); \\ Antti Karttunen, May 26 2017

From Antti Karttunen, May 26 2017: (Start)
a(n) = A062354(n) / A009223(n).
a(A000040(n)) = A084921(n). - after Enrique Pérez Herrero's May 17 2012 comment in the latter sequence.
(End)

A082054 Sum of common prime divisors (without multiplicity) of sigma(n) and phi(n).

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 5, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0, 5, 2, 2, 2, 2, 2, 0, 2, 2, 5, 0, 2, 5, 2, 2, 2, 5, 2, 2, 5, 2, 2, 2, 3, 0, 2, 2, 2, 5, 2, 5, 2, 2, 2, 2, 2, 5, 2, 0, 5, 2, 2, 2, 2, 5, 2, 3, 2, 5, 2, 2, 5, 5, 2, 2, 0, 2, 2, 2, 2, 5, 2, 7, 2, 5, 2, 2, 2, 2, 5, 2, 2, 3, 5, 0, 2, 2, 2, 5, 5

Offset: 1

Labos Elemer, Apr 03 2003

nonn

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

Cf. A000010, A000203, A008472, A009223.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; Table[Apply[Plus, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}]
a[n_] := Module[{g = GCD[DivisorSigma[1, n], EulerPhi[n]]}, If[g == 1, 0, Total[FactorInteger[g][[;; , 1]]]]]; Array[a, 100] (* Amiram Eldar, Feb 16 2025 *)

a(n)=my(f=factor(gcd(sigma(n=factor(n)), eulerphi(n)))[,1]); sum(i=1,#f,f[i]) \\ Charles R Greathouse IV, Dec 09 2013

a(n) = A008472(A009223(n)). - Amiram Eldar, Feb 16 2025

A082055 Product of common prime-divisors (without multiplicity) of sigma(n) and phi(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 6, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 1, 2, 2, 6, 1, 2, 6, 2, 2, 2, 6, 2, 2, 6, 2, 2, 2, 3, 1, 2, 2, 2, 6, 2, 6, 2, 2, 2, 2, 2, 6, 2, 1, 6, 2, 2, 2, 2, 6, 2, 3, 2, 6, 2, 2, 6, 6, 2, 2, 1, 2, 2, 2, 2, 6, 2, 10, 2, 6, 2, 2, 2, 2, 6, 2, 2, 3, 6, 1, 2, 2, 2, 6, 6

Offset: 1

Labos Elemer, Apr 03 2003

nonn

The squarefree kernel of the greatest common divisor of sigma(n) and phi(n). - Antti Karttunen, Jan 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A000010, A007947, A009223, A081396, A082054.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Apply[Times, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}]

A082055(n) = factorback(factorint(gcd(sigma(n), eulerphi(n)))[, 1]); \\ Antti Karttunen, Jan 22 2020

a(n) = A007947(A009223(n)). - Antti Karttunen, Jan 22 2020

cp's OEIS Frontend

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

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A055008 Numbers k such that gcd(phi(k), sigma(k)) = 1 with phi = A000010, sigma = A000203.

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

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A066087 a(n) = gcd(sigma(n), phi(n)) - gcd(sigma(rad(n)), phi(rad(n))); rad = A007947.

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A073815 Least number x such that gcd(phi(x), sigma(x)) = n.

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A074389 a(n) = gcd(n, sigma(n), phi(n)).

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A081396 Number of common prime factors (ignoring multiplicity) of sigma(n) = A000203(n) and phi(n) = A000010(n).

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