A001913 -id:A001913 - cp's OEIS frontend

A105874 Primes for which -2 is a primitive root.

5, 7, 13, 23, 29, 37, 47, 53, 61, 71, 79, 101, 103, 149, 167, 173, 181, 191, 197, 199, 239, 263, 269, 271, 293, 311, 317, 349, 359, 367, 373, 383, 389, 421, 461, 463, 479, 487, 503, 509, 541, 557, 599, 607, 613, 647, 653, 661, 677, 701, 709, 719, 743, 751, 757, 773, 797

Offset: 1

N. J. A. Sloane, Apr 24 2005

nonn

Also primes for which (p-1)/2 (==-1/2 mod p) is a primitive root. [Joerg Arndt, Jun 27 2011]

Joerg Arndt, Table of n, a(n) for n = 1..10000
L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
Index entries for primes by primitive root

Cf. A001122, A019334-A019338, A001913, A019339-A019367 etc., A105875-A105914.

with(numtheory); f:=proc(n) local t1,i,p; t1:=[]; for i from 1 to 500 do p:=ithprime(i); if order(n,p) = p-1 then t1:=[op(t1),p]; fi; od; t1; end; f(-2);

pr=-2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_,q_]:=Sum[2 Cos[2^n Pi/((2 q+1) (2 p+1))], {n,1,2 q p}];
Select[Range[400], Reduce[a[#, 1] == 1, Integers] &];
2 % + 1 (* Gerry Martens, Apr 28 2015 *)

forprime(p=3,10^4,if(p-1==znorder(Mod(-2,p)),print1(p", "))); /* Joerg Arndt, Jun 27 2011 */

from sympy import n_order, nextprime
from itertools import islice
def A105874_gen(startvalue=3): # generator of terms >= startvalue
    p = max(startvalue-1,2)
    while (p:=nextprime(p)):
        if n_order(-2,p) == p-1:
            yield p
A105874_list = list(islice(A105874_gen(),20)) # Chai Wah Wu, Aug 11 2023

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime belonging to this sequence when a(p,1)==1. - Gerry Martens, May 21 2015

A087026 Euler's totient of n-th cyclic number.

Original entry on oeis.org

77760, 1382399778816000, 98221093640386560, 1053590173895849103360, 821095240766161265049600000, 511883170649353472639948811682935064652736000, 410719109641406326635474611575828305116907342385175347200

Offset: 1

Reinhard Zumkeller, Jul 30 2003

nonn

Calculated with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

Max Alekseyev, Table of n, a(n) for n = 1..51 (first 42 terms from Ray Chandler)
Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A001913, A087020, A087021, A087022, A087023, A087024, A087025.

a(n) = A000010(A004042(n+1)).

a(12)-a(42) from Ray Chandler, Nov 16 2011

A021023 Decimal expansion of 1/19.

Original entry on oeis.org

0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2, 1, 0, 5, 2, 6, 3, 1, 5, 7, 8

Offset: 0

N. J. A. Sloane

The 18-digit cycle 1, 0, 5, 2, 6, 3, 1, 5, 7, 8, 9, 4, 7, 3, 6, 8, 4, 2 in this sequence and the others based on nineteenths gives the successive digits of the smallest integer which is doubled, quadrupled and octupled when the last three digits in turn are moved from the right hand end to the left hand end. For example, 842105263157894736 is eight times 105263157894736842. - Ian Duff, Jan 07 2009, Jan 12 2009
The magic square that uses the decimals of 1/19 is fully magic. 383 has the same property (see A021387). For other such primes see A072359. - Michel Marcus, Sep 02 2015
Since 19 is prime and the cycle of its reciprocal's base 10 digits is 19 - 1 long, 19 is a full reptend prime (A001913). - Alonso del Arte, Mar 21 2020

Martin Gardner, Cyclic numbers, Mathematical Circus, Chapter 10, p. 172, of the 1992 Mathematical Association of America edition.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.

Wikipedia, Prime reciprocal magic square
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,-1,1).

Cf. A021387, A072359.

Prepend[First@ RealDigits[N[1/19, 120]], 0] (* Michael De Vlieger, Sep 02 2015 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,-1,1},{0,5,2,6,3,1,5,7,8,9},100] (* or *) PadRight[{},100,{0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1}] (* Harvey P. Dale, Jan 23 2021 *)

default(realprecision,2000);1/19.0 \\ Anders Hellström, Sep 02 2015

def longDivRecip(n: Int, places: Int = 100): List[Int] = {
  val pow10 = Math.pow(10, Math.ceil(Math.log10(Math.abs(n)))).toInt
  val digits = new scala.collection.mutable.ListBuffer[Int]()
  var quotient = pow10; var remainder = 0
  while (digits.size < places) {
    remainder = quotient % n; quotient /= n; digits += quotient
    quotient = remainder * 10
  }
  digits.toList
}
0 :: longDivRecip(19) // Alonso del Arte, Mar 20 2020

G.f.: -x*(x^8 + x^7 + 2*x^6 + 4*x^5 - 2*x^4 - 3*x^3 + 4*x^2 - 3*x + 5)/((x - 1)*(x + 1)*(x^2 - x + 1)*(x^6 - x^3 + 1)). - Colin Barker, Aug 15 2012

A087020 Greatest prime factor of n-th cyclic number.

Original entry on oeis.org

37, 5882353, 333667, 513239, 121499449, 11111111111111111111111, 154083204930662557781201849, 39526741, 9999999900000001, 59779577156334533866654838281, 119968369144846370226083377, 8396862596258693901610602298557167100076327481

Offset: 1

Reinhard Zumkeller, Jul 30 2003

nonn

A004042(n) factorized with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

			A004042(2) = 142857 = 37*13*11*3^3, therefore a(1) = 37.

Max Alekseyev, Table of n, a(n) for n = 1..51 (first 42 terms from Ray Chandler)
Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number

Cf. A001913, A087021, A087022, A087023, A087024, A087025, A087026.

a(n) = A006530(A004042(n+1)).

a(12) from Ray Chandler, Nov 16 2011

A087022 Total number of prime factors of n-th cyclic number.

Original entry on oeis.org

6, 9, 12, 10, 11, 9, 11, 23, 25, 25, 22, 18, 19, 15, 14, 38, 35, 24, 28, 25, 27, 21, 17, 38, 44, 27, 43, 16, 16, 23, 42, 35, 37, 30, 29, 14, 23, 62, 41, 51, 28, 26, 24, 19, 50, 29, 39, 25, 62, 36, 29

Offset: 1

Reinhard Zumkeller, Jul 30 2003

A004042(n) factorized with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

			A004042(2) = 142857 = 37*13*11*3^3, therefore a(1) = A087021(1)+2 = 6.

Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number

Cf. A001913, A087020-A087026.

a(n) = A001222(A004042(n+1)).

a(3) corrected, a(12)-a(42) added by Ray Chandler, Nov 16 2011

a(43)-a(51) from Max Alekseyev, May 13 2022

A056619 Smallest prime with primitive root n, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 11, 2, 3, 2, 19, 2, 0, 2, 3, 2, 7, 2, 5, 2, 3, 2, 11, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 19, 2, 3, 2, 0, 2, 7, 2, 3, 2, 19, 2, 5, 2, 3, 2, 13, 2, 5, 2, 3, 2, 5, 2, 11, 2, 3, 2, 5, 2, 11, 2, 3, 2, 7, 2, 7, 2, 3, 2

Offset: 1

Robert G. Wilson v, Aug 07 2000

nonn

a(n) > n/2 for n in { 2, 6, 10, 34 }. Are there any other such indices n? - M. F. Hasler, Feb 21 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Here the primitive root may be larger than the prime, whereas in A023049 it may not be.

Cf. A001122, A001913, A019334-A019421.

f:= proc(n) local p;
   if n::odd then return 2
   elif issqr(n) then return 0
   fi;
   p:= 3;
   do
      if numtheory:-order(n,p) = p-1 then return p fi;
      p:= nextprime(p);
   od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 21 2017

a[n_] := Module[{p}, If[OddQ[n], Return[2], If[IntegerQ[Sqrt[n]], Return[0], p = 3; While[True, If[MultiplicativeOrder[n, p] == p-1, Return[p]]; p = NextPrime[p]]]]];
Array[a, 100] (* Jean-François Alcover, Apr 10 2019, after Robert Israel *)

A056619(n)=forprime(p=2,n*2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1&&return(p)) \\ or, more efficient:
A056619(n)=if(bittest(n,0),2,!issquare(n)&&forprime(p=3,n*2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1&&return(p))) \\ M. F. Hasler, Feb 21 2017

a(n) = 0 only for perfect squares, A000290.

a(n) = 2 for all odd n. a(n) = 0 for even squares. a(n) = 3 for n = 2 (mod 6). a(n) = 5 for n in {12, 18, 22, 28} (mod 30). - M. F. Hasler, Feb 21 2017

Corrected and extended by Jud McCranie, Mar 21 2002

Corrected by Robert Israel, Feb 21 2017

A060257 Numbers k such that 1/prime(k) has period prime(k) - 1.

Original entry on oeis.org

4, 7, 8, 9, 10, 15, 17, 18, 25, 29, 30, 32, 35, 39, 41, 42, 44, 48, 50, 51, 55, 56, 57, 65, 68, 73, 75, 76, 77, 81, 84, 89, 93, 94, 95, 96, 97, 100, 105, 106, 108, 114, 118, 120, 126, 127, 129, 132, 141, 142, 143, 148, 150, 154, 159, 160, 162, 164, 165, 166, 171

Offset: 1

Jeff Burch, Mar 22 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022)

Cf. A060258.

Cf. A000720 (PrimePi), A001913 (primes with primitive root 10).

Select[Range[200], MultiplicativeOrder[10, (p = Prime[#])] == p - 1 &] (* Amiram Eldar, Oct 03 2021 *)

a(n) = PrimePi(A001913(n)). - Alexander Adamchuk, Jan 28 2007

A087024 Number of divisors of n-th cyclic number.

Original entry on oeis.org

32, 384, 1280, 576, 1536, 384, 1536, 4194304, 16777216, 3538944, 1769472, 110592, 221184, 13824, 6912, 48318382080, 9663676416, 3538944, 75497472, 14155776, 56623104, 884736, 55296, 57982058496, 3710851743744, 37748736, 695784701952, 27648, 27648, 2654208

Offset: 1

Reinhard Zumkeller, Jul 30 2003

nonn

Calculated with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

Max Alekseyev, Table of n, a(n) for n = 1..51 (first 42 terms from Ray Chandler)
Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A001913, A087020, A087021, A087022, A087023, A087025, A087026.

a(n) = A000005(A004042(n+1)).

a(12)-a(42) from Ray Chandler, Nov 16 2011

A087025 Sum of divisors of n-th cyclic number.

Original entry on oeis.org

255360, 17205180696931968, 1951532915603927040, 12430490030984319840000, 9956179688423375044684800000, 6080633705911286890368486450696472902951229440, 4809378729404888733987843426081300724182624228548100710400

Offset: 1

Reinhard Zumkeller, Jul 30 2003

nonn

Calculated with Dario Alpern's ECM.

Extended using factors of 10^(A001913(n)-1)-1, see Kamada link.

Max Alekseyev, Table of n, a(n) for n = 1..51 (first 42 terms from Ray Chandler)
Dario A. Alpern, Factorization using the Elliptic Curve Method.
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Cyclic Number
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A001913, A087020, A087021, A087022, A087023, A087024, A087026.

a(n) = A000203(A004042(n+1)).

a(12)-a(42) from Ray Chandler, Nov 16 2011

A021027 Decimal expansion of 1/23.

Original entry on oeis.org

0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3, 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3, 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3, 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3, 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6

Offset: 0

N. J. A. Sloane

Since 23 is prime and the cycle of its reciprocal's base 10 digits is 22, 23 is a full reptend prime in base 10 (A001913). - Alonso del Arte, Mar 26 2020

			1/23 = 0.043478260869565217391304347826...

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A001913.

Join[{0}, RealDigits[1/23, 10, 100][[1]]] (* Alonso del Arte, Mar 14 2020 *)

def longDivRecip(n: Int, places: Int = 100): List[Int] = {
  val pow10 = Math.pow(10, Math.ceil(Math.log10(Math.abs(n)))).toInt
  val digits = new scala.collection.mutable.ListBuffer[Int]()
  var quotient = pow10; var remainder = 0
  while (digits.size < places) {
    remainder = quotient % n; quotient /= n; digits += quotient
    quotient = remainder * 10
  }
  digits.toList
}
0 :: longDivRecip(23) // Alonso del Arte, Mar 25 2020

cp's OEIS Frontend

A105874 Primes for which -2 is a primitive root.

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A087026 Euler's totient of n-th cyclic number.

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A021023 Decimal expansion of 1/19.

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A087020 Greatest prime factor of n-th cyclic number.

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A087022 Total number of prime factors of n-th cyclic number.

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A056619 Smallest prime with primitive root n, or 0 if no such prime exists.

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A060257 Numbers k such that 1/prime(k) has period prime(k) - 1.

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A087024 Number of divisors of n-th cyclic number.

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