A088144 -id:A088144 - cp's OEIS frontend

A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).

1, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, -1, 0, 0

Offset: 1

T. D. Noe, Nov 03 2003

sign

Note that A049092 lists prime(n) such that a(n) = 0. Similarly, A078330 lists prime(n) such that a(n) = -1. See A088179 for prime(n) such that a(n) = 1. Also note that a(n) == A088144(n) (mod prime(n)).

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 236.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Ed Pegg, Jr., Moebius Function (and squarefree numbers)
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A089495 (mu(p+1) for prime p), A089496 (mu(p+1)+mu(p-1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).

Cf. A049092, A078330, A088144, A088179.

[MoebiusMu(NthPrime(n)-1): n in [1..100]]; // Vincenzo Librandi, Dec 23 2018

Mathematica
```
Table[MoebiusMu[Prime[n]-1], {n, 150}]
```
PARI
```
a(n)=moebius(prime(n)-1)
```

a(n) = A067460(n) - 1. - Benoit Cloitre, Nov 04 2003

If p = prime(n), then a(n) is congruent modulo p to the sum of all primitive roots modulo p. [Uspensky and Heaslet]. - Michael Somos, Feb 16 2020

A049092 Primes p such that p-1 is not squarefree.

Original entry on oeis.org

5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 127, 137, 149, 151, 157, 163, 173, 181, 193, 197, 199, 229, 233, 241, 251, 257, 269, 271, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487

Offset: 1

Labos Elemer

nonn

Primes p with mu(p-1)=0, where mu is the Möbius function. - T. D. Noe, Nov 03 2003
Primes p such that the sum of the primitive roots of p (see A088144) is 0 mod p. - Jon Wharf, Mar 12 2015
The relative density of this sequence within the primes is 1 - A005596 = 0.626044... - Amiram Eldar, Feb 10 2021

			p = 257 is here because p-1 = 256 = 2^8.
p = 997 is here because p-1 = 996 = 3*(2^2)*83.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Moebius Function.

Cf. A005596, A039787, A078330 (primes p with mu(p-1)=-1), A088179 (primes p such that mu(p-1)=1), A089451 (mu(p-1) for prime p), A145199.

[ p: p in PrimesUpTo(500) | not IsSquarefree(p-1) ]; // Vincenzo Librandi, Mar 12 2015

Select[Prime[Range[400]], MoebiusMu[ #-1]==0&]

forprime(p=2,500,if(!issquarefree(p-1),print(p))) \\ Michael B. Porter, Mar 16 2015

a(n) = A145199(n) + 1. - Amiram Eldar, Feb 10 2021

A123475 Product of the primitive roots of prime(n).

Original entry on oeis.org

1, 2, 6, 15, 672, 924, 11642400, 163800, 109681110000, 5590307923200, 970377408, 134088514560000, 138960660963091968000, 874927557504000, 3456156426256013065185600000000, 30688148115024695887527936000000

Offset: 1

T. D. Noe, Sep 27 2006

nonn

Except for n=2, we have a(n)=1 (mod prime(n)).

			a(5)=672 because the primitive roots of 11 are {2,6,7,8}.

C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

Charles R Greathouse IV, Table of n, a(n) for n = 1..145

Cf. A060749 (primitive roots of prime(n)), A088144 (sum of primitive roots of prime(n)).

PrimRoots[p_] := Select[Range[p-1], MultiplicativeOrder[ #,p]==p-1&]; Table[Times@@PrimRoots[Prime[n]], {n,20}]
Times@@@Table[PrimitiveRootList[Prime[n]], {n, 20}] (* Harlan J. Brothers, Sep 02 2023 *)

vecprod(v)=prod(i=1,#v,v[i])
a(n,p=prime(n))=vecprod(select(n->znorder(Mod(n,p))==p-1,[2..p-1]))
apply(p->a(0,p), primes(20)) \\ Charles R Greathouse IV, May 15 2015

use ntheory ":all"; sub list { my $n=shift; grep { znorder($,$n) == $n-1 } 2..$n-1; } say vecprod(list($)) for @{primes(nth_prime(20))}; # Dana Jacobsen, May 15 2015

A266987 Primes p for which the average of the primitive roots equals p/2.

Original entry on oeis.org

2, 5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

From Robert Israel, Feb 01 2016: (Start)
Union of A002144 and A267010.
Contains A002144 because for each of these primes, x is a primitive root iff p-x is a primitive root. (End)

			a(13) = 13 since the primitive roots of 13 are 2, 6, 7, and 11 and the average of these primitive roots is (2+6+7+11)/phi(12) = 26/4 = 13/2.

Robert Israel, Table of n, a(n) for n = 1..8213

Cf. A002144, A008330, A060749, A088144, A266986, A267010.

proots := proc(n)
    local r,eulphi,m;
    if n = 1 then
        return {0} ;
    end if;
    eulphi := numtheory[phi](n) ;
    r := {} ;
    for m from 0 to n-1 do
        if numtheory[order](m,n) = eulphi then
            r := r union {m} ;
        end if;
    end do:
    return r;
end proc:
isA266987 := proc(n)
    local r;
    if isprime(n) then
        r := convert(proots(n),list) ;
        2*add(pr, pr=r)  = n*nops(r) ;
    else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA266987(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
Filter:= proc(p) local x, s,js;
  if p mod 4 = 1 then return true fi;
  x:= numtheory:-primroot(p);
  js:= select(t -> igcd(t,p-1)=1, [$1..p-2]);
  s:= add(x&^ j mod p, j=js);
  evalb(s = p/2 * nops(js))
end proc:
select(Filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 01 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 100}]; Prime[Flatten[Position[A, _?(# == 1 &)]]]
(* second program (version >= 10): *)
Select[Prime[Range[100]], Mean[PrimitiveRootList[#]] == #/2&] (* Jean-François Alcover, Jan 12 2016 *)

a(n) = prime(A266986(n)).

A121380 Sums of primitive roots for n (or 0 if n has no primitive roots).

Original entry on oeis.org

0, 1, 2, 3, 5, 5, 8, 0, 7, 10, 23, 0, 26, 8, 0, 0, 68, 16, 57, 0, 0, 56, 139, 0, 100, 52, 75, 0, 174, 0, 123, 0, 0, 136, 0, 0, 222, 114, 0, 0, 328, 0, 257, 0, 0, 208, 612, 0, 300, 200, 0, 0, 636, 156, 0, 0, 0, 348, 886, 0, 488, 216, 0, 0, 0, 0, 669, 0, 0, 0

Offset: 1

Ed Pegg Jr, Jul 25 2006

In Article 81 of his Disquisitiones Arithmeticae (1801), Gauss proves that the sum of all primitive roots (A001918) of a prime p, mod p, equals MoebiusMu[p-1] (A008683). "The sum of all primitive roots is either = 0 (mod p) (when p-1 is divisible by a square), or = +-1 (mod p) (when p-1 is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)."

			The primitive roots of 13 are 2, 6, 7, 11. Their sum is 26, or 0 (mod 13). By Gauss, 13-1=12 is thus divisible by a square number.

J. C. F. Gauss, Disquisitiones Arithmeticae, 1801.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein, Primitive Roots.

Cf. A001918, A008683, A046147 (primitive roots of n), A088144, A088145, A123475, A222009.

primitiveRoots[n_] := If[n == 1, {}, If[n == 2, {1}, Select[Range[2, n-1], MultiplicativeOrder[#, n] == EulerPhi[n] &]]]; Table[Total[primitiveRoots[n]], {n,100}]
(* From version 10 up: *)
Table[Total @ PrimitiveRootList[n], {n, 1, 100}] (* Jean-François Alcover, Oct 31 2016 *)

A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 2, 8, 7, 6, 2, 6, 11, 7, 3, 10, 5, 11, 14, 7, 12, 6, 2, 13, 14, 15, 3, 10, 5, 10, 20, 17, 11, 21, 19, 15, 7, 14, 2, 8, 3, 19, 18, 14, 27, 21, 26, 10, 11, 15, 3, 17, 13, 24, 22, 12, 11, 21, 2, 32, 17, 13, 15, 18, 35, 5, 20, 24, 22, 19

Offset: 1

Geoffrey Critzer, May 03 2015

Each row is a complete set of incongruent primitive roots.
Each row is a permutation of the corresponding row in A060749.
Row lengths are A008330.
T(n,1) = A001918(n).

			1;
2;
2,  3;
3,  5;
2,  8,  7,  6;
2,  6, 11,  7;
3, 10,  5, 11, 14,  7, 12,  6;
2, 13, 14, 15,  3, 10;
5, 10, 20, 17, 11, 21, 19, 15,  7, 14;
2,  8,  3, 19, 18, 14, 27, 21, 26, 10, 11, 15;
Row 6 contains 2,6,11,7 because 13 is the 6th prime number. 2 is the least positive primitive root of 13. The integers relatively prime to 13-1=12 are {1,5,7,11}. So we have: 2^1==2, 2^5==6, 2^7==11, and 2^11==7 (mod 13).

Alois P. Heinz, Rows n = 1..120, flattened

Cf. A001918, A008330, A060749.
Last elements of rows give A255367.
Row sums give A088144.

with(numtheory):
T:= n-> (p-> seq(primroot(p)&^k mod p, k=select(
         h-> igcd(h, p-1)=1, [$1..p-1])))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, May 03 2015

Table[nn = p;Table[Mod[PrimitiveRoot[nn]^k, nn], {k,Select[Range[nn - 1], CoprimeQ[#, nn - 1] &]}], {p,Prime[Range[12]]}] // Grid

A266986 The indices of primes p for which the average of the primitive roots equals p/2.

Original entry on oeis.org

1, 3, 6, 7, 8, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 63, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135, 136, 137, 139, 140, 142, 145

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

The average of the primitive roots of a prime p are <,=, or > p/2 (observation).
The indices of all primes p==1(mod 4) are in this sequence since for primes of form 4k+1 b a primitive root implies -b a primitive root.
The indices of some primes p==3 (mod 4) are also in this sequence although for most such primes the average of the primitive roots is <> p/2.(observation)

			p(a(1))=p(1)=2. 2 has the primitive root 1. The average primitive root is 1 and 1=2/2.
p(a(2))=p(3)=5. The primitive roots of 5 are 2 and 3. Their average equals (2+3)/phi(4)=5/2=p/2.

Dimitri Papadopoulos, Table of n, a(n) for n = 1..505

Cf. A008330, A060749, A088144, A266987.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Flatten[Position[A, _?(# == 1 &)]]

A266989 Primes for which the average of the primitive roots is < p/2.

Original entry on oeis.org

31, 43, 67, 223, 379, 491, 619, 631, 643, 683, 859, 883, 907, 1051, 1091, 1423, 1747, 1987, 2143, 2347, 2371, 2467, 2531, 2767, 3307, 3643, 3691, 3739, 3823, 3931, 4019, 4219, 4519, 4691, 4987, 5059, 5107, 5347, 5683, 5827, 6043

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

These primes are congruent to 3 (mod 4).

			a(1)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A008330, A060749, A088144, A266987, A266988, A267010.

f:= proc(p) local g;
  if not isprime(p) then return false fi;
  g:= numtheory[primroot](p);
  evalb(add(g&^i mod p, i = select(t->igcd(t,p-1)=1, [$1..p-2]))
     < p/2 * numtheory:-phi(p-1))
end proc:
select(f, [seq(i,i=3..10000,4)]); # Robert Israel, Feb 09 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1,100}]; Prime[Flatten[Position[A, _?(# < 1 &)]]]

ar(p) = my(r, pr, j); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsort(r) ;
isok(p) = my(vr = ar(p)); vecsum(vr)/#vr < p/2;
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Feb 09 2016

a(n) = prime(A266988(n)).

A266990 The indices of primes p for which the average of the primitive roots is > p/2.

Original entry on oeis.org

2, 4, 5, 9, 15, 17, 20, 22, 23, 27, 28, 31, 32, 34, 36, 38, 39, 41, 43, 46, 47, 49, 52, 54, 56, 58, 61, 64, 67, 69, 72, 73, 76, 81, 83, 85, 86, 90, 91, 92, 93, 95, 96, 99, 101, 103, 105, 107, 109, 111, 118, 120, 125, 128, 129, 131, 132, 133, 138, 141, 143, 144, 146, 150

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

It appears that these primes are all congruent to 3 (mod 4).

			a(2) = 4 is a term since prime(a(2)) = prime(4) = 7, the primitive roots of 7 are 3 and 5 and their average is (3+5)/2 = 8/2 > 7/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A008330, A060749, A088144, A267009.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Flatten[Position[A, _?(# > 1 &)]]
Select[Range[150], Mean[PrimitiveRootList[(p = Prime[#])]] > p/2 &] (* Amiram Eldar, Oct 09 2021 *)

a(n) = A000720(A267009(n)). - Amiram Eldar, Oct 09 2021

A267009 Primes p for which the average of the primitive roots is > p/2.

Original entry on oeis.org

3, 7, 11, 23, 47, 59, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 227, 239, 251, 263, 271, 283, 311, 331, 347, 359, 367, 383, 419, 431, 439, 443, 463, 467, 479, 487, 499, 503, 523, 547, 563, 571, 587, 599, 607, 647, 659, 691, 719, 727

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

It appears that these primes are all congruent to 3 (mod 4).

			a(2) = 7 since the primitive roots of 7 are 3 and 5 and their average is (3+5)/2 = 8/2 > 7/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008330, A060749, A088144, A266990.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}], Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Prime[Flatten[Position[A, _?(# > 1 &)]]]
Select[Range[1000], PrimeQ[#] && Mean[PrimitiveRootList[#]] > #/2 &] (* Amiram Eldar, Oct 09 2021 *)

a(n) = prime(A266990(n)).

cp's OEIS Frontend

A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).

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A049092 Primes p such that p-1 is not squarefree.

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A123475 Product of the primitive roots of prime(n).

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A266987 Primes p for which the average of the primitive roots equals p/2.

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Maple

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A121380 Sums of primitive roots for n (or 0 if n has no primitive roots).

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A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

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Maple

Mathematica

A266986 The indices of primes p for which the average of the primitive roots equals p/2.

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