A010554 -id:A010554 - cp's OEIS frontend

A378508 Values taken by phi(phi(m)) (A010554).

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 72, 80, 82, 84, 88, 92, 96, 100, 104, 108, 112, 120, 128, 130, 132, 144, 156, 160, 162, 164, 168, 172, 176, 178, 180, 184, 190, 192, 200, 204, 208, 212, 216, 220, 224, 232, 238, 240, 250, 252, 256, 260, 264, 272, 276, 280, 288, 292, 300, 312, 320, 324, 328, 336, 344, 348, 352, 356, 358, 360, 368, 380, 384, 396, 400

Offset: 1

Amiram Eldar, Nov 29 2024

nonn

Numbers k such that A378506(k) > 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.

Cf. A000010, A010554, A378506.

Subsequence of A002202.

q[k_] := AnyTrue[PhiInverse[k], PhiMultiplicity[#] > 0 &]; Select[Range[1000], q] (* using David M. Bressoud's CNT.m *)

is(k) = {my(v = invphi(k)); for(i = 1, #v, if(istotient(v[i]), return(1))); 0;} \\ using Max Alekseyev's invphi.gp

A046144 Number of primitive roots modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 4, 0, 4, 2, 0, 0, 8, 2, 6, 0, 0, 4, 10, 0, 8, 4, 6, 0, 12, 0, 8, 0, 0, 8, 0, 0, 12, 6, 0, 0, 16, 0, 12, 0, 0, 10, 22, 0, 12, 8, 0, 0, 24, 6, 0, 0, 0, 12, 28, 0, 16, 8, 0, 0, 0, 0, 20, 0, 0, 0, 24, 0, 24, 12, 0, 0, 0, 0, 24, 0, 18, 16, 40, 0, 0, 12, 0, 0, 40, 0, 0

Offset: 1

Eric W. Weisstein

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000
S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A010554, A033949, A046145, A046146, A008330, A002233, A071894, A219027.

A046144 := proc(n)
    local a,eulphi,m;
    if n = 1 then
        return 1;
    end if;
    eulphi := numtheory[phi](n) ;
    a := 0 ;
    for m from 0 to n-1 do
        if numtheory[order](m,n) = eulphi then
            a := a + 1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jan 12 2016

Prepend[ Table[ If[ IntegerQ[ PrimitiveRoot[n]] , EulerPhi[ EulerPhi[n]], 0], {n, 2, 91}],1] (* Jean-François Alcover, Sep 13 2011 *)

for(i=1, 100, p=0; for(q=1, i, if(gcd(q,i)==1 && znorder(Mod(q,i)) == eulerphi(i), p++)); print1(p, ", ")) /* V. Raman, Nov 22 2012 */

a(n) = my(s=znstar(n)); if(#(s.cyc)>1, 0, eulerphi(s.no)) \\ Jeppe Stig Nielsen, Oct 18 2019

use ntheory ":all"; my @A = map { !defined znprimroot($) ? 0 : euler_phi(euler_phi($)); } 0..10000; say "$ $A[$]" for 1..$#A; # Dana Jacobsen, Apr 28 2017

a(n) is equal to A010554(n) unless n is a term of A033949, in which case a(n)=0.

A046145 Smallest primitive root modulo n, or 0 if no root exists.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, 0, 0, 0, 3, 0, 2, 7, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 5, 3, 0, 0

Offset: 0

Eric W. Weisstein

The value 0 at index 0 says 0 has no primitive roots, but the 0 at index 1 says 1 has a primitive root of 0, the only real 0 in the sequence.

a(n) is nonzero if and only if n is 2, 4, or of the form p^k, or 2*p^k where p is an odd prime and k>0. - Tom Edgar, Jun 02 2014

T. D. Noe, Table of n, a(n) for n = 0..10000
Pēteris K. Siliņš, Cross ratios for finite field geometries, Bachelor's Thesis, Univ. Groningen (Netherlands, 2024). See p. 17.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046146, A002233, A071894, A219027, A008330, A010554, A111076, A285512, A285513, A285514.

A046145 := proc(n)
  if n <=1 then
    0;
  else
    pr := numtheory[primroot](n) ;
    if pr = FAIL then
       return 0 ;
    else
       return pr ;
    end if;
  end if;
end proc:
seq(A046145(n),n=0..110) ;  # R. J. Mathar, Jul 08 2010

smallestPrimitiveRoot[n_ /; n <= 1] = 0; smallestPrimitiveRoot[n_] := Block[{pr = PrimitiveRoot[n], g}, If[! NumericQ[pr], g = 0, g = 1; While[g <= pr, If[ CoprimeQ[g, n] && MultiplicativeOrder[g, n] == EulerPhi[n], Break[]]; g++]]; g]; smallestPrimitiveRoot /@ Range[0, 100] (* Jean-François Alcover, Feb 15 2012 *)
f[n_] := Block[{pr = PrimitiveRootList[n]}, If[pr == {}, 0, pr[[1]]]]; Array[f, 105, 0] (* v10.0 Robert G. Wilson v, Nov 04 2014 *)

{ A046145(n) = for(q=1,n-1, if(gcd(q,n)==1 && znorder(Mod(q,n))==eulerphi(n), return(q);)); 0; } /* V. Raman, Nov 22 2012, edited by Max Alekseyev, Apr 20 2017 */

use ntheory ":all"; say "$ ", znprimroot($) || 0  for 0..100; # Dana Jacobsen, Mar 16 2017

Initial terms corrected by Harry J. Smith, Jan 27 2005

A046146 Largest primitive root modulo n, or 0 if no root exists.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 5, 5, 0, 5, 7, 8, 0, 11, 5, 0, 0, 14, 11, 15, 0, 0, 19, 21, 0, 23, 19, 23, 0, 27, 0, 24, 0, 0, 31, 0, 0, 35, 33, 0, 0, 35, 0, 34, 0, 0, 43, 45, 0, 47, 47, 0, 0, 51, 47, 0, 0, 0, 55, 56, 0, 59, 55, 0, 0, 0, 0, 63, 0, 0, 0, 69, 0, 68, 69, 0, 0, 0, 0, 77, 0, 77, 75, 80, 0, 0

Offset: 0

Eric W. Weisstein

nonn

The value 0 at index 0 says 0 has no primitive roots, but the 0 at index 1 says 1 has a primitive root of 0, the only real 0 in the sequence. - Initial terms corrected by Harry J. Smith, Jan 27 2005

a(n) is nonzero if and only if n is 2, 4, or of the form p^k, or 2*p^k where p is an odd prime and k>0. - Tom Edgar, Jun 02 2014

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A002233, A071894, A219027, A008330, A010554.

f[n_] := Block[{pr = PrimitiveRootList[n]}, If[pr == {}, 0, pr[[-1]]]]; Array[f, 86, 0] (* Robert G. Wilson v, Nov 03 2014 *)

for(i=0,100,p=0;for(q=1,i-1,if(gcd(q,i)==1&&znorder(Mod(q,i))==eulerphi(i),p=q));print1(p",")) /* V. Raman, Nov 22 2012 */

Initial terms corrected by Harry J. Smith, Jan 27 2005

A111725 Number of residues modulo n of the maximum order.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 2, 4, 4, 8, 2, 6, 4, 6, 4, 10, 7, 8, 4, 6, 6, 12, 4, 8, 8, 12, 8, 8, 6, 12, 6, 8, 8, 16, 6, 12, 12, 8, 10, 22, 8, 12, 8, 16, 8, 24, 6, 16, 14, 18, 12, 28, 8, 16, 8, 24, 16, 24, 12, 20, 16, 30, 8, 24, 14, 24, 12, 16, 18, 24, 8, 24, 24, 18, 16, 40, 14, 32, 12

Offset: 1

Max Alekseyev, Nov 18 2005

nonn

The maximum order modulo n is given by A002322(n).
a(n) is the number of primitive lambda-roots of n. - Michel Marcus, Mar 17 2016
A primitive lambda-root is an element of maximal order modulo n. - Joerg Arndt, Mar 19 2016
a(n) is odd if and only if n is a factor of 24, i.e., n is in A018253. - Jianing Song, Apr 27 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009.
P. J. Cameron and D. A. Preece, Primitive lambda-roots, 2014.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.
S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Act. Arithm. 86 (2) (1998) 113, Theorem 2.1

Cf. A002322, A008330, A300064, A300065, A300079, A300080.

LiDelta := proc(q,n)
    local a,p,e,lam,v ;
    a := 0 ;
    lam := numtheory[lambda](n) ;
    for p in numtheory[factorset](n) do
        e := padic[ordp](n,p) ;
        if p =2 and e= 3 and q =2 and padic[ordp](lam,q) = 1 then
            return A083399(n) ;
        elif isprime(q) then
            v := padic[ordp](lam,q) ;
            if modp( numtheory[lambda](p^e),q^v) = 0 then
                a := a+1 ;
            end if;
        end if:
    end do:
    a ;
end proc:
A111725 := proc(n)
    local a,q ;
    a := 1;
    for q in numtheory[factorset](numtheory[lambda](n)) do
        a := a*(1-1/q^LiDelta(q,n)) ;
    end do:
    a*numtheory[phi](n) ;
end proc:
seq(A111725(n),n=1..30) ; # R. J. Mathar, Sep 29 2017

f[list_]:=Count[list,Max[list]];Map[f,Table[Table[MultiplicativeOrder[k,n],{k,Select[Range[n],GCD[#,n]==1&]}],{n,1,100}]]  (* Geoffrey Critzer, Jan 26 2013 *)

{ a(n) = my(r, c, r1); r=1; c=0; for(k=0, n-1, if(gcd(k, n)!=1, next); r1=znorder(Mod(k,n)); if(r1==r, c++); if(r1>r, r=r1; c=1) ); c; }

{ A111725(n) = if(n<3,return(1)); my(k,p); k=znstar(n)[2]; p=factor(k[1])[,1]; eulerphi(n) * prod(i=1,#p, (1-1/p[i]^vecsum(apply(x->valuation(k[1]\x,p[i])==0,k))) ); } \\ Max Alekseyev, Oct 23 2021

For prime n, a(n) = phi(phi(n)) = A010554(n) = phi(n-1). - Nick Hobson, Jan 09 2007
Decompose (Z/nZ)* as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) = Sum_{d divides psi(n)} (mu(psi(n)/d)*Product{i=1..m} gcd(d, k_i)). This is an immediate corollary from the fact that the number of elements in (Z/nZ)* such that x^d == 1 (mod n) is Product{i=1..m} gcd(d, k_i). Here (Z/nZ)* is the multiplicative group of integers modulo n, psi(n) = A002322(n) and mu(n) = A008683(n). - Jianing Song, Apr 27 2019
From Jianing Song, Oct 12 2021: (Start)
Decompose (Z/nZ)* as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) = phi(n) * Product_{p prime dividing phi(n)} (1 - 1/p^r(p)), where r(p) is the number of j such that v(k_j,p) = v(k_m,p), v(s,p) is the p-adic valuation of s.
Proof: let G = (Z/nZ)*, G_p be the Sylow p-subgroup of G, then G = Product_{p prime dividing phi(n)} G_p: every element x can be uniquely written as Product_{p prime dividing phi(n)} x_p for x_p in G_p. Let ord(x) be the order of x. Since ord(x_p, x_p') = 1 for distinct p and p', we have ord(x) = Product_{p prime dividing phi(n)} ord(x_p), hence x is of maximal order if and only if each x_p is of maximal order in G_p.
Each G_p is isomorphic to C_{p^(e_1)} x C_{p^(e_2)} x ... x C_{p^(e_m)} for e_1 <= e_2 <= ... <= e_m, e_m > 0. Write x_p = (x_{p,1}, x_{p,2}, ..., x_{p,m}). Suppose that e_m = e_{m-1} = ... = e_{m-r+1} > e_{m-r}, then x_p is of maximal order in G_p if and only of x_{p,j} is of order p^(e_m) for some m-r+1 <= j <= m, so the number of such x_p is p^(e_1) * p^(e_2) * ... * p^(e_{m-r}) * (p^(r*e_m) - p^(r*((e_m)-1))) = |G_p| * (1 - 1/p^r).
An example: n = 15903, then (Z/nZ)* = C_6 x C_18 x C_90. We can see that r(2) = 3, r(3) = 2 and r(5) = 1, so a(15903) = phi(15903) * (1 - 1/2^3) * (1 - 1/3^2) * (1 - 1/5^1) = 6048.
It should be clear that a(n) = phi(phi(n)) if and only if r(p) = 1 for every prime p dividing phi(n), or v(k_{m-1},p) < v(k_m,p) for every prime p dividing phi(n). Otherwise, a(n) > phi(phi(n)). (End)

A049107 a(n) = Euler phi function applied 5 times to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 4, 2, 4, 2, 2

Offset: 1

Labos Elemer

nonn

			For n = 163, the successive iterates applying Euler totient function are as follows: 163, 162, 54, 18, 6, 2, 1. The 6th term is 2, when Phi was applied 5 times. So a(163)=2, already a power of 2.
For n = 487, the successive iterates are 486, 162, 54, 18, 6, 2, 1. On the fifth iteration we reach 6, thus a(487) = 6. This is also the first term of A049107 that is not a power of 2. - _Antti Karttunen_, Aug 22 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A010554, A049099, A049100, A049115.

a(n)=Nest[ EulerPhi, n, 5 ]
Nest[EulerPhi,Range[110],5] (* Harvey P. Dale, May 19 2019 *)

A049107(n) = eulerphi(eulerphi(eulerphi(eulerphi(eulerphi(n))))); \\ Antti Karttunen, Aug 22 2017

a(n) = A000010(A049100(n)) = A010554(A049099(n)) = phi(phi(phi(phi(phi(n))))), where phi = A000010. - Antti Karttunen, Aug 22 2017

A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

Original entry on oeis.org

1, 3, 6, 7, 12, 9, 16, 15, 18, 17, 28, 19, 32, 23, 30, 31, 48, 27, 46, 35, 40, 39, 62, 39, 60, 45, 54, 47, 76, 45, 76, 63, 68, 65, 74, 55, 92, 65, 78, 71, 112, 61, 104, 79, 84, 85, 132, 79, 110, 85, 114, 91, 144, 81, 126, 95, 112, 105, 164, 91, 152, 107, 118, 127, 144, 101

Offset: 1

Labos Elemer, Jan 14 2000

nonn

For n = 2^w, the sum is 2^(w+1) - 1.

			If phi is applied repeatedly to n = 91, the iterates {91, 72, 24, 8, 4, 2, 1} are obtained. Their sum is a(91) = 91 + 72 + 24 + 8 + 4 + 2 + 1 = 202.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A010554, A049099, A049100, A049107, A049108, A032358.

Row sums of A375478.

a053478 = (+ 1) . sum . takeWhile (/= 1) . iterate a000010
-- Reinhard Zumkeller, Oct 27 2011

f[n_] := Plus @@ Drop[ FixedPointList[ EulerPhi, n], -1]; Table[ f[n], {n, 66}] (* Robert G. Wilson v, Dec 16 2004 *)
f[1] := 1; f[n_] := n + f[EulerPhi[n]]; Table[f[n], {n, 66}] (* Carlos Eduardo Olivieri, May 26 2015 *)

a(n)=my(s=n);while(n>1,s+=n=eulerphi(n)); s \\ Charles R Greathouse IV, Feb 21 2013

a(n) = n + a(phi(n)).

a(n) = A092693(n) + n. - Vladeta Jovovic, Jul 02 2004

A049099 a(n) = Euler phi function applied thrice to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 4, 8, 2, 4, 4, 4, 4, 10, 4, 4, 4, 8, 4, 8, 2, 8, 4, 4, 4, 12, 4, 8, 4, 4, 8, 8, 4, 8, 8, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 8, 6, 8, 16, 4, 16, 4, 8, 8, 16, 4, 8, 8, 8, 10, 8, 8, 16, 4, 8, 8, 16

Offset: 1

Labos Elemer

nonn

			n=163: the successive iterates of Euler totient function are 163,162,54,18,6,2,1. The 4th term is 18, when Phi was applied 3 times. So a(163)=18.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A000010, A010554, A049100, A049107, A049115.

Mathematica
```
a(n)=Nest[ EulerPhi, n, 3 ]
```

A049099(n) = eulerphi(eulerphi(eulerphi(n))); \\ Antti Karttunen, Aug 22 2017

a(n) = A000010(A010554(n)) = A000010(A000010(A000010(n))). - Antti Karttunen, Aug 22 2017

A049100 a(n) = Euler phi function applied 4 times to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 1, 4, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 4, 8, 2, 8, 2, 4, 4, 8, 2, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 8, 4, 8, 4, 4

Offset: 1

Labos Elemer

nonn

			n=163, the successive iterates applying Euler totient function are as follows: 163,162,54,18,6,2,1. The 5th term is 6, when Phi was applied 4 times. So a(163)=6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A000010, A010554, A049099, A049107, A049115.

with(numtheory): seq(phi(phi(phi(phi(n)))),n=1..130); # Emeric Deutsch, May 14 2006

Mathematica
```
a(n)=Nest[ EulerPhi, n, 4 ]
```

A049100(n) = eulerphi(eulerphi(eulerphi(eulerphi(n)))); \\ Antti Karttunen, Aug 22 2017

a(n) = phi(phi(phi(phi(n)))) = A000010(A000010(A000010(A000010(n)))) = A010554(A010554(n)) = A000010(A049099(n)).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 23 2007

A256608 Longest eventual period of a^(2^k) mod n for all a.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 2, 6, 1, 2, 4, 10, 1, 4, 2, 6, 2, 3, 1, 4, 1, 4, 1, 2, 2, 6, 6, 2, 1, 4, 2, 6, 4, 2, 10, 11, 1, 6, 4, 1, 2, 12, 6, 4, 2, 6, 3, 28, 1, 4, 4, 2, 1, 2, 4, 10, 1, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 1, 18, 4, 20, 2, 1, 6

Offset: 1

Ivan Neretin, Apr 04 2015

nonn

a(n) is a divisor of phi(phi(n)) (A010554).

			In other words, eventual period of {0..n-1} under the map x -> x^2 mod n.
For example, with n=10 the said map acts as follows. Read down the columns: the column headed 2 for example means that (repeatedly squaring mod 10), 2 goes to 4 goes to 16 = 6 (mod 10) goes to 36 = 6 mod 10 --- and has reached a fixed point.
0 1 2 3 4 5 6 7 8 9
0 1 4 9 6 5 6 9 4 1
0 1 6 1 6 5 6 1 6 1
0 1 6 1 6 5 6 1 6 1
and thus every number reaches a fixed point. This means the eventual common period is 1, hence a(10)=1.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A001146, A002322, A002326, A007733, A010554, A037178.
First differs from A256607 at n=43.
LCM of entries in row n of A279185.

a[n_] := With[{lambda = CarmichaelLambda[n]}, MultiplicativeOrder[2, lambda / (2^IntegerExponent[lambda, 2])]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2016 *)

rpsi(n) = lcm(znstar(n)[2]); \\ A002322
pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733
a(n) = pb(rpsi(n)); \\ Michel Marcus, Jan 28 2016

a(n) = A007733(A002322(n)).

a(prime(n)) = A037178(n). - Michel Marcus, Jan 27 2016

Name changed by Jianing Song, Feb 02 2025

cp's OEIS Frontend

A378508 Values taken by phi(phi(m)) (A010554).

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A046144 Number of primitive roots modulo n.

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A046145 Smallest primitive root modulo n, or 0 if no root exists.

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A046146 Largest primitive root modulo n, or 0 if no root exists.

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A111725 Number of residues modulo n of the maximum order.

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A049107 a(n) = Euler phi function applied 5 times to n.

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A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

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Crossrefs