A046144 -id:A046144 - cp's OEIS frontend

A380604 Numbers k such that there is no number i such that A046144(i) = 2*k.

7, 13, 15, 17, 19, 21, 23, 25, 28, 29, 31, 33, 34, 35, 37, 38, 39, 43, 45, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 83, 85, 87, 91, 92, 93, 94, 97, 98, 99, 101, 103, 104, 105, 107, 109, 111, 112, 113, 114, 115, 117, 118

Offset: 1

David James Sycamore, Jan 28 2025

nonn

2*a(n) are the even numbers which are not in A378508, namely numbers 2*m for which no number exists which has 2*m primitive roots. See A380594 for discussion of even numbers which are not in this sequence.

			 There is no x such that A046144(x) = 14, so 7 is a term in this sequence (see also A380594).

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. A046144, A378508, A380594.

q[n_] := Count[Join @@ PhiInverse[PhiInverse[2*n]], ?(IntegerQ[PrimitiveRoot[#]] &)] == 0; Select[Range[120], q] (* _Amiram Eldar, Jan 28 2025, using David M. Bressoud's CNT.m *)

isA033948(n) = {my(f = factor(n)); lcm(znstar(f)[2]) == eulerphi(f); }
isok(k) = {my(v = invphi(2*k), w, c = 0); for(i = 1, #v, c += vecsum(apply(x -> isA033948(x), invphi(v[i])))); c == 0; } \\ Amiram Eldar, Jan 28 2025, using Max Alekseyev's invphi.gp

A379883 a(1) = 1. Let j = a(n-1) and r = A046144(j). Then for n > 1, if j is novel and r > 0, a(n) = r. If j is novel and r = 0 then a(n) = 1. If j has occurred k (>1) times already then a(n) = k*j.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 4, 8, 1, 6, 1, 7, 2, 6, 12, 1, 8, 16, 1, 9, 2, 8, 24, 1, 10, 2, 10, 20, 1, 11, 4, 12, 24, 48, 1, 12, 36, 1, 13, 4, 16, 32, 1, 14, 2, 12, 48, 96, 1, 15, 1, 16, 48, 144, 1, 17, 8, 32, 64, 1, 18, 2, 14, 28, 1, 19, 6, 18, 36, 72, 1, 20, 40, 1, 21, 1, 22, 4, 20, 60, 1, 23, 10, 30, 1, 24, 72, 144, 288, 1, 25, 8, 40, 80, 1, 26, 4

Offset: 1

David James Sycamore, Jan 09 2025

In other words if j = a(n-1) has not occurred earlier and has r (> 0) primitive roots then a(n) = r. Cases where novel A046144(j) = 0 cannot be counted multiplicatively (as k*j) for repeats, so a(n) = 1 is designed to permit the sequence to continue past such points, which means including in the count of 1's terms following (1,2,3,4,6), for which it is true that r = 1. Terms beyond a(12) = 8 which count the number of 1's (by the second condition) give the cardinality of terms with no primitive roots, plus the few (5) cases of terms with primitive root = 1.

Every even number m in A380594 appears finitely many times, consequent to occasions of integers v (>6) for which A046144(v) = m, and to repetitions (k*j) = m for j even. However every odd number appears once only (consequent to odd counts of 1's). The odd numbers appear in order, and since 2 precedes all of them, the primes are in order.

			a(2) = 1 since a(1)=1 and and 1 has one primitive root. Since 1 has been seen twice, a(3) = 2 and then a(4) = 1 since 2 is a novel term with one primitive root.
a(9) = 5, a novel term with two primitive roots so a(10) = 2, which has appeared once before (a(3)=2), so a(11) = 4, the second occurrence of 4 so a(12) = 8, a novel term with no primitive roots, meaning that a(13) = 1. The count of 1's is now 6, so a(14) = 6, meaning 5 prior terms with one primitive root and one with none.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A046144, A378508, A380594, A380604.

nn = 120; c[_] := 0; j = 1;
f[x_] := f[x] = Which[
  x == 1, 1,
  IntegerQ[PrimitiveRoot[x]], Nest[EulerPhi, x, 2],
  True, 0];
{j}~Join~Reap[Monitor[Do[
  If[c[j] == 0,
    Set[k, # + Boole[# == 0]] &[f[j]]; c[j]++,
    k = ++c[j]*j ];
j = Sow[k], {n, 2, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Jan 09 2025 *)

a(78)=1 inserted by David Radcliffe, Aug 03 2025

A010554 a(n) = phi(phi(n)), where phi is the Euler totient function.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

If n has a primitive root, then it has exactly phi(phi(n)) of them (Burton 1989, p. 188), which means that if p is a prime number, then there are exactly phi(p-1) incongruent primitive roots of p (Burton 1989). - Jonathan Vos Post, Sep 10 2010

See A046144 for the number of primitive roots mod n. - Wolfdieter Lang, Mar 09 2012

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Burton, D. M. "The Order of an Integer Modulo n," "Primitive Roots for Primes," and "Composite Numbers Having Primitive Roots." Sections 8.1-8.3 in Elementary Number Theory, 4th ed. Dubuque, IA: William C. Brown Publishers, pp. 184-205, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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]
S. R. Finch, Idempotents and Nilpotents Modulo n (arXiv:math.NT/0605019)
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A000010, A049099, A049100, A049107, A077197.

a010554 = a000010 . a000010  -- Reinhard Zumkeller, Dec 26 2012

[EulerPhi(EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Feb 24 2018

Maple
```
with(numtheory): f := n->phi(phi(n));
```

Table[EulerPhi[EulerPhi[n]],{n,0,200}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)
Nest[EulerPhi[#]&,Range[100],2] (* Harvey P. Dale, Jan 13 2024 *)

a(n)=eulerphi(eulerphi(n)) \\ Charles R Greathouse IV, Feb 06 2017

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

A046147 Triangle read by rows in which row n lists the primitive roots mod n (omitting numbers n without a primitive root).

Original entry on oeis.org

1, 2, 3, 2, 3, 5, 3, 5, 2, 5, 3, 7, 2, 6, 7, 8, 2, 6, 7, 11, 3, 5, 3, 5, 6, 7, 10, 11, 12, 14, 5, 11, 2, 3, 10, 13, 14, 15, 7, 13, 17, 19, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 2, 3, 8, 12, 13, 17, 22, 23, 7, 11, 15, 19, 2, 5, 11, 14, 20, 23, 2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26

Offset: 2

Eric W. Weisstein

			n followed by primitive roots, if any:
1 -
2 1
3 2
4 3
5 2 3
6 5
7 3 5
8 -
9 2 5
10 3 7
11 2 6 7 8
12 -
13 2 6 7 11
...

T. D. Noe, Table of n, a(n) for n = 2..3119 (first 99 nonempty rows of triangle, flattened)
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144 (row lengths), A046145, A046146.

Cf. A060749, A306252 (1st column), A306253 (last/maximum element)

f:= proc(n) local p,k,m,R;
     p:= numtheory:-primroot(n);
     if p = FAIL then return NULL fi;
     m:= numtheory:-phi(n);
     k:= select(i -> igcd(i,m) = 1, [$1..m-1]);
     op(sort(map(t -> p&^t mod n, k)))
end proc:
f(2):= 1:
map(f, [$2..50]); # Robert Israel, Apr 28 2017

a[n_] := Select[Range[n-1], GCD[#, n] == 1 && MultiplicativeOrder[#, n] == EulerPhi[n]& ]; Table[a[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Oct 23 2012 *)
PrimitiveRootList[Range[Prime[10]]]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 10 2016 *)

a_row(r) = my(v=[], phi=eulerphi(r)); for(i=1, r-1, if(1 == gcd(r, i) && phi == znorder(Mod(i, r)), v=concat(v, i))); v \\ Ruud H.G. van Tol, Oct 23 2023

Edited by Robert Israel, Apr 28 2017

A103309 Smallest prime primitive root of n that is less than n, or 0 if none exists.

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, Jan 29 2005

Differs from A046145 only for indices n = 2, 41, 109, 151, 229, ...; see A103335. - Jeppe Stig Nielsen, Mar 06 2020

Robert Israel, Table of n, a(n) for n = 0..10000
G. Martin, The Least Prime Primitive Root and the Shifted Sieve, Acta Arith. 80 (1997), no. 3, 277-288; arXiv:math/9807104 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103310, A103335.

F:= proc(n)
  local r;
  r:= numtheory:-primroot(n);
  while r::integer and not isprime(r) do
    r:= numtheory:-primroot(r,n);
  od:
  if r = FAIL then 0 else r fi
end proc:
seq(F(n),n=0..200); # Robert Israel, May 18 2015

a[n_] := SelectFirst[PrimitiveRootList[n], PrimeQ[#] && # < n&] /. Missing["NotFound"] -> 0;
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 15 2017 *)

A103310 Largest prime primitive root of n that is less than n, or 0 if none exists.

Original entry on oeis.org

0, 0, 0, 2, 3, 3, 5, 5, 0, 5, 7, 7, 0, 11, 5, 0, 0, 11, 11, 13, 0, 0, 19, 19, 0, 23, 19, 23, 0, 19, 0, 17, 0, 0, 31, 0, 0, 19, 29, 0, 0, 29, 0, 29, 0, 0, 43, 43, 0, 47, 47, 0, 0, 41, 47, 0, 0, 0, 47, 47, 0, 59, 53, 0, 0, 0, 0, 61, 0, 0, 0, 67, 0, 59, 61, 0, 0, 0, 0, 59, 0, 59, 71, 79, 0, 0, 73

Offset: 0

Harry J. Smith, Jan 29 2005

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309.

hasproot:= proc(n)
  if n::odd then nops(numtheory:-factorset(n))=1
  else padic:-ordp(n,2)=1 and nops(numtheory:-factorset(n/2))=1
  fi
end proc;
hasproot(2):= true: hasproot(4):= true:
f:= proc(n) local p,t;
  if not hasproot(n) then return 0 fi;
  t:= numtheory:-phi(n);
  p:= prevprime(n);
  while not numtheory:-order(p,n)=t do
    if p = 2 then return 0 fi;
    p:= prevprime(p);
  od;
  p
end proc:
f(0):= 0: f(1):= 0: f(2):= 0:
map(f, [$0..100]); # Robert Israel, Sep 08 2020

a[n_] := Module[{R = PrimitiveRootList[n], s}, s = Select[R, # < n && PrimeQ[#]&]; If[s == {}, 0, s[[-1]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 01 2023 *)

A103335 Numbers whose smallest primitive root (A046145) is not prime.

Original entry on oeis.org

1, 2, 41, 109, 151, 229, 251, 271, 313, 337, 362, 367, 409, 439, 542, 626, 674, 733, 761, 818, 878, 971, 991, 1021, 1031, 1069, 1289, 1297, 1303, 1429, 1471, 1489, 1681, 1759, 1783, 1789, 1811, 1871, 1873, 1879, 2062, 2137, 2342, 2411, 2441, 2551, 2594

Offset: 1

Harry J. Smith, Jan 31 2005

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A001918, A046144, A046145, A046146, A103309, A103310, A103336, A103337, A103338.

filter:= proc(n)
  local r;
  r:= numtheory:-primroot(n);
  r <> FAIL and not isprime(r)
end proc:
filter(1):= true:
select(filter, [$1..3000]); Robert Israel, Sep 08 2020

L = {}; Do[ If[!PrimeQ[ Min[ Select[ Range[n], CoprimeQ[#, n] && MultiplicativeOrder[#, n] == CarmichaelLambda[n] &]]],
L = Append[L, n]], {n, 1, 3000}]; L (* Jonathan Sondow, May 17 2017 *)

Offset changed by Robert Israel, Sep 08 2020

A219027 Number of non-primitive roots for n, less than n.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 4, 7, 6, 7, 6, 11, 8, 11, 14, 15, 8, 15, 12, 19, 20, 17, 12, 23, 16, 21, 20, 27, 16, 29, 22, 31, 32, 25, 34, 35, 24, 31, 38, 39, 24, 41, 30, 43, 44, 35, 24, 47, 36, 41, 50, 51, 28, 47, 54, 55, 56, 45, 30, 59, 44, 53, 62, 63, 64, 65, 46, 67, 68

Offset: 1

V. Raman, Nov 10 2012

nonn

a(n) will be the same as A219029(n) except when n is a member of A033949 or n = 1, i.e. n is not 2, 4, prime, power of a prime, twice a prime, or twice a prime power. In such cases, when n is a member of A033949, then a(n) = n-1.

Cf. A008330 (number of primitive roots for the n-th prime, less than n-th prime).
Cf. A046144 (number of primitive roots for n, less than n).
Cf. A010554 (value of phi(phi(n))).
Cf. A219029.

for(i=1,100,p=0;for(q=1,i-1,if(gcd(q,i)>1||znorder(Mod(q,i))!=eulerphi(i),p++));print1(p","))

n-1-A046144(n).

cp's OEIS Frontend

A380604 Numbers k such that there is no number i such that A046144(i) = 2*k.

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A379883 a(1) = 1. Let j = a(n-1) and r = A046144(j). Then for n > 1, if j is novel and r > 0, a(n) = r. If j is novel and r = 0 then a(n) = 1. If j has occurred k (>1) times already then a(n) = k*j.

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A010554 a(n) = phi(phi(n)), where phi is the Euler totient function.

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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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A046147 Triangle read by rows in which row n lists the primitive roots mod n (omitting numbers n without a primitive root).

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Maple

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PARI

Extensions

A103309 Smallest prime primitive root of n that is less than n, or 0 if none exists.

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