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A384490 Numbers m such that both roots of x^2 - x - 1 modulo m are primitive roots modulo m.

41, 61, 109, 149, 241, 269, 389, 409, 449, 569, 601, 641, 701, 821, 929, 1129, 1181, 1201, 1301, 1321, 1429, 1481, 1489, 1609, 1801, 1889, 1901, 1949, 2129, 2141, 2309, 2341, 2381, 2549, 2609, 2741, 2909, 3061, 3109, 3181, 3209, 3221, 3229, 3361, 3449, 3541

Offset: 1

Jay Anderson, May 31 2025

nonn

Empirical observation: For each m in this sequence A001175(m) = m-1 and A015134(m) = m+2.

			For m = 41 the roots of x^2 - x - 1 (mod 41) are 7 and 35. 7 and 35 are both primitive roots modulo 41.

Wikipedia, Primitive root modulo n.

Cf. A001175, A015134.

test[p_]:=Module[{inv2,sqr},If[JacobiSymbol[5,p]==1,inv2=ModularInverse[2,p]; sqr=PowerMod[5,1/2,p]; {MultiplicativeOrder[Mod[inv2*(sqr-1),p],p],MultiplicativeOrder[Mod[inv2*(-sqr-1),p],p]} == {p-1,p-1},False]]; Cases[Prime[Range[4, 5000]], ?(test[#] &)] (* _Shenghui Yang, Jun 01 2025 *)

{ forprime(p=2, 3600, s=polrootsmod(x^2 - x - 1,p);
 if( #s==2 && p-1==znorder(Mod(s[1],p)) && p-1==znorder(Mod(s[2],p)),
 print1(p,", "); ); ); } \\ Joerg Arndt, May 31 2025

A365519 Moebius inversion of A015134.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 3, 2, 13, 4, 6, 3, 8, 8, 8, 9, 21, 8, 24, 13, 11, 16, 10, 6, 9, 12, 62, 8, 33, 16, 24, 24, 24, 36, 18, 63, 24, 32, 42, 24, 21, 48, 24, 33, 69, 64, 21, 10, 32, 24, 26, 27, 144, 48, 40, 62, 61, 32, 62, 99, 72, 32, 48, 24, 33, 96, 88, 24

Offset: 1

Jay Anderson, Sep 07 2023

nonn

			For n=4, the a(4) = 2 cycles counted are
  {0 1 1 2 3 1} repeating
  {0 3 3 2 1 3} repeating
There are 2 other cycles {0} and {0 2 2} mod 4 which are not counted because they are multiples of cycles counted at earlier n (n=1 for {0} and n=2 for {0 1 1}*2 = {0 2 2}).

Cf. A008683, A015134.

a[n_]:=Sum[MoebiusMu[d]Part[ResourceFunction["OEISSequence"]["A015134"],n/d],{d,Divisors[n]}]; Array[a,70] (* Stefano Spezia, Sep 27 2023 *)

a(n) = Sum_{d|n} moebius(d)*A015134(n/d).

A015134(n) = Sum_{d|n} a(d).

A238596 Number of distinct sequences defined by the upper left value in powers of n X n (0,1) matrices.

Original entry on oeis.org

2, 6, 50, 1140, 86052

Offset: 1

Jay Anderson, Mar 01 2014

A sequence can be defined by powers of a matrix with only 0 and 1 values. For instance, the upper left value in the matrix M^n where M=[0 1; 1 1] is the Fibonacci sequence.

Also, the number of distinct sequences defined by all element values in powers of n X n (0, 1) matrices (see A239748) that start with 1. - Christopher Hunt Gribble, May 12 2014

			a(2) = 6 since there are 6 distinct sequences for 2 X 2 (0,1) matrices:
[0 0; 0 0] => 0 0 0 0 0 ...
[1 0; 0 0] => 1 1 1 1 1 ...
[0 1; 0 0] => 0 0 0 0 0 ...
[1 1; 0 0] => 1 1 1 1 1 ...
[0 0; 1 0] => 0 0 0 0 0 ...
[1 0; 1 0] => 1 1 1 1 1 ...
[0 1; 1 0] => 0 1 0 1 0 ...
[1 1; 1 0] => 1 2 3 5 8 ...
[0 0; 0 1] => 0 0 0 0 0 ...
[1 0; 0 1] => 1 1 1 1 1 ...
[0 1; 0 1] => 0 0 0 0 0 ...
[1 1; 0 1] => 1 1 1 1 1 ...
[0 0; 1 1] => 0 0 0 0 0 ...
[1 0; 1 1] => 1 1 1 1 1 ...
[0 1; 1 1] => 0 1 1 2 3 ...
[1 1; 1 1] => 1 2 4 8 16 ...

Cf. A239748.

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User: Jay Anderson

A384490 Numbers m such that both roots of x^2 - x - 1 modulo m are primitive roots modulo m.

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A365519 Moebius inversion of A015134.

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A238596 Number of distinct sequences defined by the upper left value in powers of n X n (0,1) matrices.

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