A046147 - OEIS

%I #34 Nov 25 2023 20:57:26

%S 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,

%T 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,

%U 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

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

%H T. D. Noe, <a href="/A046147/b046147.txt">Table of n, a(n) for n = 2..3119</a> (first 99 nonempty rows of triangle, flattened)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimitiveRoot.html">Primitive Root.</a>

%e n followed by primitive roots, if any:

%e 1 -

%e 2 1

%e 3 2

%e 4 3

%e 5 2 3

%e 6 5

%e 7 3 5

%e 8 -

%e 9 2 5

%e 10 3 7

%e 11 2 6 7 8

%e 12 -

%e 13 2 6 7 11

%e ...

%p f:= proc(n) local p,k,m,R;

%p      p:= numtheory:-primroot(n);

%p      if p = FAIL then return NULL fi;

%p      m:= numtheory:-phi(n);

%p      k:= select(i -> igcd(i,m) = 1, [$1..m-1]);

%p      op(sort(map(t -> p&^t mod n, k)))

%p end proc:

%p f(2):= 1:

%p map(f, [$2..50]); # _Robert Israel_, Apr 28 2017

%t 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 *)

%t PrimitiveRootList[Range[Prime[10]]]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 10 2016 *)

%o (PARI) 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

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

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

%K nonn,tabf

%O 2,2

%A _Eric W. Weisstein_

%E Edited by _Robert Israel_, Apr 28 2017