A008330 -id:A008330 - cp's OEIS frontend

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

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)).

A286714 Difference phi(p-1) - phi(p+1) for lesser of twin primes.

Original entry on oeis.org

-1, 0, 0, 2, 4, 4, 12, 0, 8, 16, 20, 32, 40, 8, 24, 40, 32, 60, 4, 24, 60, 84, 24, 56, 24, 136, 104, 36, 44, 116, 184, 48, 84, 184, 68, 252, 72, 280, 68, 144, 56, 292, 140, 192, 120, 338, 276, 120, 144, 262, 192, 376, 120, 268, 192, 236, 64, 168, 240, 492, 348, 388

Offset: 1

Michel Marcus, May 13 2017

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stephan Ramon Garcia, Elvis Kahoro, Florian Luca, Primitive root discrepancy for twin primes, arXiv:1705.02485 [math.NT], 2017.

Cf. A001359, A006512, A008330, A286715.

(EulerPhi[#-1] - EulerPhi[#+1]) &@ Select[Prime@ Range@ 310, PrimeQ[# + 2] &] (* Giovanni Resta, May 13 2017 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+2), print1(eulerphi(p-1)-eulerphi(p+1), ", ")));

A286715 Lesser of twin primes for which phi(p-1) < phi(p+1).

Original entry on oeis.org

3, 2381, 3851, 14561, 17291, 20021, 20231, 26951, 34511, 41231, 47741, 50051, 52361, 55931, 57191, 65171, 67211, 67271, 70841, 82811, 87011, 98561, 101501, 101531, 108461, 117041, 119771, 126491, 129221, 134681, 136991, 142871, 145601, 150221, 156941, 165551, 166601

Offset: 1

Michel Marcus, May 13 2017

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stephan Ramon Garcia, Elvis Kahoro, Florian Luca, Primitive root discrepancy for twin primes, arXiv:1705.02485 [math.NT], 2017.

Cf. A001359, A006512, A008330, A286714.

s = {}; p = 3; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; If[EulerPhi[p + 1] > EulerPhi[p - 1], AppendTo[s, p]]; p = q, {15500}]; s (* Amiram Eldar, Sep 11 2019 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+2) && (eulerphi(p-1) < eulerphi(p+1)), print1(p, ", ")));

A104039 Number of primitive roots modulo prime(n)^2, where prime(n) is n-th prime.

Original entry on oeis.org

1, 2, 8, 12, 40, 48, 128, 108, 220, 336, 240, 432, 640, 504, 1012, 1248, 1624, 960, 1320, 1680, 1728, 1872, 3280, 3520, 3072, 4000, 3264, 5512, 3888, 5376, 4536, 6240, 8704, 6072, 10656, 6000, 7488, 8748, 13612, 14448, 15664, 8640, 13680, 12288, 16464

Offset: 1

Lekraj Beedassy, Mar 31 2005

nonn

I. Niven, H. S. Zuckerman & H. L. Montgomery, An Introduction to the Theory of Numbers, 5th Ed., p. 102, John Wiley, NY, 1991.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[(NthPrime(n)-1)*EulerPhi((NthPrime(n)-1)): n in [1..50]]; // Vincenzo Librandi, Aug 18 2017

with(numtheory): for p from 1 to 100 do printf(`%d,`,(ithprime(p)-1)*phi(ithprime(p)-1)) od: # James Sellers, Apr 10 2005

Table[(Prime[n] - 1) EulerPhi[(Prime[n] - 1)], {n, 50}] (* Vincenzo Librandi, Aug 18 2017 *)

a(n) = (prime(n) - 1)*phi((prime(n) - 1)) = A006093(n)*A000010(A006093(n)) = A006093(n)*A008330(n).

More terms from James Sellers, Apr 10 2005

A121519 Number of primitive roots of prime A103203(n).

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 16, 22, 24, 28, 40, 52, 64, 72, 82, 84, 88, 112, 128, 130, 132, 144, 156, 172, 178, 190, 192, 232, 238, 250, 252, 276, 280, 292, 324, 358, 384, 396, 400, 418, 424, 430, 442, 448, 480, 490, 508, 520, 544, 552, 592, 612, 640, 652, 658, 682, 712

Offset: 1

Klaus Brockhaus, Aug 06 2006

nonn

Also records in A008330.

			Prime A103203(5) = 23 has 10 primitive roots, so a(5) = 10; 23 = prime(9) and A008330(9) = 10 is a record in A008330.

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

Cf. A008330, A103203.

r=0;for(n=1,230,k=eulerphi(prime(n)-1);if(r

A174842 Irregular triangle T(i,n) giving the number of elements of Zp having multiplicative order di, the i-th divisor of p-1, where p is the n-th prime.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Mar 30 2010

The divisors of p-1 are assumed to be in increasing order. The first row, for prime 2, has only one term. All other rows begin with two 1s and end with phi(p-1). There are tau(p-1), the number of divisors of p-1, terms in each row. The sum of the terms in each row is p-1. When p is a prime of the form 4k-1, then the last two terms in the row are equal. When p is a prime of the form 4k+1, then the last two terms in the row have a ratio of 2.

			For prime p=17, the 7th prime, the multiplicative order of the numbers 1 to p-1 is 1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2. There is one 1, one 2, two 4's, four 8's, and eight 16's. Hence row 7 is 1, 1, 2, 4, 8.

T. D. Noe, Rows n=1..500 of triangle, flattened
Eric W. Weisstein, MathWorld: Modulo Multiplication Group

A008328 (tau(p-1)), A008330 (phi(p-1)), A174843 (divisors of p-1)

Flatten[Table[EulerPhi[Divisors[p-1]], {p, Prime[Range[100]]}]]

T(i,n) = phi(di), where di is the i-th divisor of prime(n)-1.

A219029 a(n) = n - 1 - phi(phi(n)).

Original entry on oeis.org

-1, 0, 1, 2, 2, 4, 4, 5, 6, 7, 6, 9, 8, 11, 10, 11, 8, 15, 12, 15, 16, 17, 12, 19, 16, 21, 20, 23, 16, 25, 22, 23, 24, 25, 26, 31, 24, 31, 30, 31, 24, 37, 30, 35, 36, 35, 24, 39, 36, 41, 34, 43, 28, 47, 38, 47, 44, 45, 30, 51, 44, 53, 50, 47, 48, 57, 46, 51, 48

Offset: 1

V. Raman, Nov 10 2012

sign

There are exactly n - 1 - phi(phi(n)) non-primitive roots for n, less than n, if n is prime.

a(n) will be the same as A219027(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.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008330 (number of primitive roots for the n-th prime).
Cf. A046144 (number of primitive roots for n).
Cf. A010554 (value of phi(phi(n))).
Cf. A219027, A219428.

[(n - 1 - EulerPhi(EulerPhi(n))): n in [1..70] ]; // Vincenzo Librandi, Jan 26 2013

Table[n - (EulerPhi[EulerPhi[n]] + 1), {n, 75}] (* Alonso del Arte, Nov 17 2012 *)

for(n=1,100,print1(n-1-eulerphi(eulerphi(n))","))

a(n) = n - 1 - A010554(n). - V. Raman, Nov 22 2012

A241197 Numerator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 1, 1, 4, 8, 16, 288, 256, 192, 768, 384, 3456, 3072, 6912, 6144, 55296, 1658880, 221184, 110592, 3317760, 442368, 13271040

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The denominator is in A241198.

			In decimal, the minima are about 1, 0.5, 0.333333, 0.266667, 0.228571, 0.207792, 0.196856, 0.195569, 0.191808, 0.185194, 0.183469, 0.181713, 0.180525, 0.173812, 0.172676, 0.171024, 0.165507, 0.165127, 0.163588.

Cf. A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Numerator[Transpose[tMin][[2]]]

a(20)-a(22) from Giovanni Resta, Apr 14 2016

A241198 Denominator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

Original entry on oeis.org

1, 2, 3, 15, 35, 77, 1463, 1309, 1001, 4147, 2093, 19019, 17017, 39767, 35581, 323323, 10023013, 1339481, 676039, 20957209, 2800733, 86822723

Offset: 1

T. D. Noe, Apr 17 2014

The values of p are in A241196. The numerator is in A241197.

Cf. A008330 (phi(prime(n)-1)), A073918, A241194, A241195.

tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Denominator[Transpose[tMin][[2]]]

a(20)-a(22) from Giovanni Resta, Apr 14 2016

A247176 Largest number of maximal order mod n.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 7, 5, 7, 8, 11, 11, 5, 13, 13, 14, 11, 15, 17, 19, 19, 21, 23, 23, 19, 23, 23, 27, 23, 24, 29, 29, 31, 33, 31, 35, 33, 37, 37, 35, 31, 34, 41, 43, 43, 45, 43, 47, 47, 46, 45, 51, 47, 53, 53, 53, 55, 56, 53, 59, 55, 61, 61, 63, 61, 63, 65, 67, 67, 69, 67

Offset: 1

Eric Chen, Nov 29 2014

			a(18) = 11 because the largest possible order mod 18 is 6, and because 16, 15, 14, and 12 are not coprime to 18, and the orders of 17 and 13 to mod 18 are 2 and 3, not the largest possible order, and the order of 11 to mod 18 is 6, so a(18) = 11.

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A002322 (orders), same as A046146 for n with primitive roots, A071894 (for primes).

Cf. A111076, A111725, A046145, A046144, A001918, A008330.

prms={}; f[n_] = Block[If[MultiplicativeOrder[p, n]=CarmichaelLambda[n], Join[prms, p]]; prms[-1]]; Array[f, 128]

carmichaellambda(n)=lcm(znstar(n)[2]);
for(i=1, 128, p=0; for(q=1, i-1, if(gcd(q, i)==1&&znorder(Mod(q, i))==carmichaellambda(i), p=q)); print1(p", "))

a(68) corrected by Eric Chen, Jun 01 2015

cp's OEIS Frontend

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

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A286714 Difference phi(p-1) - phi(p+1) for lesser of twin primes.

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A286715 Lesser of twin primes for which phi(p-1) < phi(p+1).

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A104039 Number of primitive roots modulo prime(n)^2, where prime(n) is n-th prime.

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A121519 Number of primitive roots of prime A103203(n).

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A174842 Irregular triangle T(i,n) giving the number of elements of Zp having multiplicative order di, the i-th divisor of p-1, where p is the n-th prime.

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A219029 a(n) = n - 1 - phi(phi(n)).

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A241197 Numerator of new minima of phi(p-1)/(p-1), where phi is Euler's totient function and p = prime(n).

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