A112927 -id:A112927 - cp's OEIS frontend

A014664 Order of 2 modulo the n-th prime.

2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316

Offset: 2

N. J. A. Sloane

In other words, a(n), n >= 2, is the least k such that prime(n) divides 2^k-1.
Concerning the complexity of computing this sequence, see for example Bach and Shallit, p. 115, exercise 8.
Also A002326((p_n-1)/2). Conjecture: If p_n is not a Wieferich prime (1093, 3511, ...) then A002326(((p_n)^k-1)/2) = a(n)*(p_n)^(k-1). - Vladimir Shevelev, May 26 2008
If for distinct i,j,...,k we have a(i)=a(j)=...=a(k) then the number N = p_i*p_j*...*p_k is in A001262 and moreover A137576((N-1)/2) = N. For example, a(16)=a(37)=a(255)=52. Therefore we could take N = p_16*p_37*p_255 = 53*157*1613 = 13421773. - Vladimir Shevelev, Jun 14 2008
Also degree of the irreducible polynomial factors for the polynomial (x^p+1)/(x+1) over GF(2), where p is the n-th prime. - V. Raman, Oct 04 2012
Is this the same as the smallest k > 1 not already in the sequence such that p = prime(n) is a factor of 2^k-1 (A270600)? If the answer is yes, is the sequence a permutation of the positive integers > 1? - Felix Fröhlich, Feb 21 2016. Answer: No, it is easy to prove that 6 is missing and obviously 11 appears twice. - N. J. A. Sloane, Feb 21 2016
pi(A112927(m)) is the index at which a given number m first appears in this sequence. - M. F. Hasler, Feb 21 2016

			2^2 == 1 (mod 3) and so a(2) = 2;
2^4 == 1 (mod 5) and so a(3) = 4;
2^3 == 1 (mod 7) and so a(4) = 3;
2^10 == 1 (mod 11) and so a(5) = 10; etc.
[Conway & Guy, p. 166]: Referring to the work of Euler, 1/13 in base 2 = 0.000100111011...; (cycle length of 12). - _Gary W. Adamson_, Aug 22 2009

E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I.
Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966; Table 48, page 98, "Exponents to Which a Belongs, MOD p and MOD p^n.
John H. Conway and Richard Guy, "The Book of Numbers", Springer-Verlag, 1996; p. 166: "How does the Cycle Length Change with the Base?". [From Gary W. Adamson, Aug 22 2009]
S. K. Sehgal, Group rings, pp. 455-541 in Handbook of Algebra, Vol. 3, Elsevier, 2003; see p. 493.

Michael De Vlieger, Table of n, a(n) for n = 2..10001 (terms 2..1001 from Nick Hobson, terms 1002..5001 from Zak Seidov)
Gary W. Adamson, Further comments.
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, Funct. Anal. Other Math. 1 (2006), 175-180.
O. N. Karpenkov, On examples of difference operators for {0,1}-valued functions over finite sets, arXiv:math/0611940 [math.CO], 2006.

Cf. A002326 (order of 2 mod 2n+1), A001122 (full reptend primes in base 2), A065941, A112927.

In other bases: A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.

P:=Filtered([1..350],IsPrime);; a:=List([2..Length(P)],n->OrderMod(2,P[n]));; Print(a); # Muniru A Asiru, Jan 29 2019

with(numtheory): [ seq(order(2,ithprime(n)), n=2..60) ];

Reap[Do[p=Prime[i];Do[If[PowerMod[2,k,p]==1,Print[{i,k}];Sow[{i,k}];Goto[ni]],{k,1,10^6}];Label[ni],{i,2,5001}]][[2,1]] (* Zak Seidov, Jan 26 2009 *)
Table[MultiplicativeOrder[2, Prime[n]], {n, 2, 70}] (* Jean-François Alcover, Dec 10 2015 *)

a(n)=if(n<0,0,k=1;while((2^k-1)%prime(n)>0,k++);k)

A014664(n)=znorder(Mod(2, prime(n))) \\ Nick Hobson, Jan 08 2007, edited by M. F. Hasler, Feb 21 2016

forprime(p=3, 800, print(factormod((x^p+1)/(x+1), 2, 1)[1, 1])) \\ V. Raman, Oct 04 2012

from sympy import n_order, prime
def A014664(n): return n_order(2,prime(n)) # Chai Wah Wu, Nov 09 2023

a(n) = (A000040(n)-1)/A001917(n); a(A072190(n)) = A001122(n) - 1. - Benoit Cloitre, Jun 06 2004

More terms from Benoit Cloitre, Apr 11 2003

A007138 Smallest primitive factor of 10^n - 1. Also smallest prime p such that 1/p has repeating decimal expansion of period n.

Original entry on oeis.org

3, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 99990001, 21401, 859, 757, 29, 3191, 211, 2791, 353, 67, 103, 71, 999999000001, 2028119, 909090909090909091

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, Jud McCranie

In the 18th century, the Japanese mathematician Ajima Naonobu (a.k.a. Ajima Chokuyen) gave the first 16 terms (Smith and Mikami, p. 199). - Jonathan Sondow, May 25 2013
Also the least prime number p such that the multiplicative order of 10 modulo p is n. - Robert G. Wilson v, Dec 09 2013
n always divides p-1. - Jon Perry, Nov 02 2014

			a(3) = 37 since 1/37 = 0.027027... has period 3, and 37 is the smallest such prime (in fact, the only one).

Ajima Naonobu (aka Ajima Chokuyen), Fujin Isshũ (Periods of Decimal Fractions).
J. Brillhart et al., Factorizations of b^n +/- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 1..438 (terms 1..364 from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Torbjörn Granlund, Factors of 10^n-1.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Koide, Factors of Repunit Numbers.
David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; chapter X.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Decimal Expansion
Wikipedia, Repeating decimals
Robert G. Wilson v and Max Alekseyev, Smallest primitive factor of 10^n -1, or 0 if not yet found, for a(n) and n=1..10000
Index entries for sequences related to decimal expansion of 1/n

First column of A046107.
Cf. A001913.
Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

S:= {}:
for n from 1 to 60 do
  F:= numtheory:-factorset(10^n-1) minus S;
  A[n]:= min(F);
  S:= S union F;
od:
seq(A[n],n=1..60); # Robert Israel, Nov 10 2014

s={}; Reap[Scan[(x=Complement[FactorInteger[10^#-1][[All,1]],s]; Sow[Min[x]]; s=Union[s,x])&,Range@60]][[2,1]] (* Shenghui Yang, Apr 15 2025 *)

b-file truncated to 364 terms as a(365) was wrong and is currently unknown (pointed by Eric Chen), and a-file revised by Max Alekseyev, Apr 26 2022

A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 3, 16, 1, 5, 5, 8, 1, 24, 1, 38, 9, 11, 3, 68, 6, 5, 4, 54, 7, 79, 1, 16, 11, 5, 13, 462, 3, 5, 13, 140, 3, 123, 7, 110, 54, 11, 7, 664, 2, 114, 29, 118, 7, 124, 59, 188, 13, 55, 3, 4456, 1, 5, 82, 96, 5, 353, 3, 118, 11, 485, 7

Offset: 1

Vladeta Jovovic, Feb 04 2001

nonn

Also, number of primitive factors of 2^n - 1 (cf. A212953). - Max Alekseyev, May 03 2022
The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). See A002326.
a(n) is odd iff n is squarefree, A005117. - Thomas Ordowski, Jan 18 2014
The set S for which a(n) = |S| contains an odd number of prime powers p^k, where k > 0 and p == 3 (mod 4), iff n is squarefree and greater than one. - Isaac Saffold, Dec 28 2019

			a(3) = |{7}| = 1, a(4) = |{5,15}| = 2, a(6) = |{9,21,63}| = 3.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 200 terms from Alois P. Heinz)

Column k=2 of A212957.
Primitive factors of b^n - 1: this sequence (b=2), A059885 (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A001037, A046801, A058943, A059912, A112927, A212953.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(2^d-1), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, May 31 2012

a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 2^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 71} ] (* Jean-François Alcover, Dec 12 2012 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(2^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} A008683(n/d) * A046801(d) = Sum_{d|A007947(n)} A008683(d) * A046801(n/d). - Max Alekseyev, May 03 2022

a(n) = 1 iff 2^n-1 is noncomposite. a(prime(n)) = 2^A088863(n)-1. - Thomas Ordowski, Jan 16 2014

More terms from John W. Layman, Mar 22 2002

More terms from Alois P. Heinz, May 31 2012

A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.

Original entry on oeis.org

2, 3, 31, 13, 11, 7, 19531, 313, 19, 521, 12207031, 601, 305175781, 29, 181, 17, 409, 5167, 191, 41, 379, 23, 8971, 390001, 101, 5227, 109, 234750601, 59, 61, 1861, 2593, 199, 3061, 211, 37, 149, 761, 79, 241, 2238236249, 43, 1644512641, 89, 1171, 47

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

Max Alekseyev, Table of n, a(n) for n = 1..520

Cf. A064081, A218357.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

p = 2; t = Table[0, {100}]; While[p < 3000000001, a = MultiplicativeOrder[5, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p]; p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

a(23)-a(40) from Robert G. Wilson v, Oct 13 2014

a(41)-a(46) from Robert G. Wilson v, Oct 15 2014

A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

2, 1, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 73, 797161, 547, 4561, 17, 1871, 19, 1597, 1181, 368089, 67, 47, 6481, 8951, 398581, 109, 29, 59, 31, 683, 21523361, 2413941289, 103, 71, 530713, 13097927, 2851, 313, 42521761, 83, 43, 431, 5501, 181, 23535794707

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064079(n).

Max Alekseyev, Table of n, a(n) for n = 1..730 (first 153 terms from Robert G. Wilson v)

Cf. A002326, A064079, A112092, A218356, A057958, A074477.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

a:= proc(n) local f,p;
f:= numtheory:-factorset(3^n - 1);
for  p in f do
   if numtheory:-order(3,p) = n then return p fi
od:
1
end proc:
seq(a(n),n=1..100); # Robert Israel, Oct 13 2014

p = 2; t = Table[0, {100}]; While[p < 100000001, a = MultiplicativeOrder[3, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]];  p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

More terms from Robert G. Wilson v, Dec 11 2013

A112092 a(n) is the least prime such that the multiplicative order of 4 mod a(n) equals n.

Original entry on oeis.org

3, 5, 7, 17, 11, 13, 43, 257, 19, 41, 23, 241, 2731, 29, 151, 65537, 43691, 37, 174763, 61681, 337, 397, 47, 97, 251, 53, 87211, 15790321, 59, 61, 715827883, 641, 67, 137, 71, 433, 223, 229, 79, 4278255361, 83, 1429, 431, 353, 631, 277, 283, 193, 4363953127297

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

a(n) is the minimal prime divisor of A064080(n).

Max Alekseyev, Table of n, a(n) for n = 1..1207 (first 100 terms from Amiram Eldar)

Cf. A064080, A112927.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

a[n_] := Module[{f = FactorInteger[4^n - 1][[;; , 1]]}, Do[p = f[[k]]; If[ MultiplicativeOrder[4, p] == n, Break[] ], {k, 1, Length[f]}]; p]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)

a(n) = {my(p = 3); while (znorder(Mod(4, p)) != n, p = nextprime(p+1)); p;} \\ Michel Marcus, Feb 08 2016

a(29)-a(30) from Michel Marcus, Feb 08 2016

More term from Amiram Eldar, Jan 27 2019

A252170 Smallest primitive prime factor of 12^n-1.

Original entry on oeis.org

11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, 3933841, 3307

Offset: 1

Eric Chen, Dec 15 2014

nonn

Also, smallest prime p such that 1/p has duodecimal period n.

			a(4) = 5 because 1/5 = 0.249724972497... and 5 is the smallest prime with period 4 in base 12.
a(5) = 22621 because 1/22621 = 0.0000100001... and 22621 is the smallest (in fact, the only one) prime with period 5 in base 12.

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Cf. A246004, A246489.

S:= {}:
for n from 1 to 72 do
  F:= numtheory:-factorset(12^n-1) minus S;
  A[n]:= min(F);
  S:= S union F;
od:
seq(A[n], n=1..72);

prms={}; Table[f=First/@FactorInteger[12^n-1]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 72}]

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(12^n-1)[,1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "););} \\ Michel Marcus, Dec 15 2014; after A007138

Edited by Max Alekseyev, Aug 26 2021

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A379640 Smallest primitive prime factor of 7^n-1.

Original entry on oeis.org

2, 1, 19, 5, 2801, 43, 29, 1201, 37, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 117307, 419, 281, 11898664849, 23, 47, 73, 2551, 53, 109, 13564461457, 59, 6568801, 311, 353, 3631, 29078814248401, 2127431041, 13841169553, 223, 351121, 486643, 41, 83, 51031

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

Also, smallest prime p such that 1/p has septimal period n.

Max Alekseyev, Table of n, a(n) for n = 1..430 (first 388 terms from Sean A. Irvine)

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Cf. A074249.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(7^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A379641 Smallest primitive prime factor of 8^n-1.

Original entry on oeis.org

7, 3, 73, 5, 31, 19, 127, 17, 262657, 11, 23, 37, 79, 43, 631, 97, 103, 87211, 32377, 41, 92737, 67, 47, 433, 601, 2731, 2593, 29, 233, 18837001, 2147483647, 193, 199, 307, 71, 246241, 223, 571, 937, 61681, 13367, 77158673929, 431, 397, 271, 139, 2351, 577

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

Also, smallest prime p such that 1/p has octal period n.

Max Alekseyev, Table of n, a(n) for n = 1..548 (first 500 terms from Sean A. Irvine)

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Cf. A274908.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(8^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

cp's OEIS Frontend

A014664 Order of 2 modulo the n-th prime.

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A007138 Smallest primitive factor of 10^n - 1. Also smallest prime p such that 1/p has repeating decimal expansion of period n.

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A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

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A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.

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A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

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A112092 a(n) is the least prime such that the multiplicative order of 4 mod a(n) equals n.

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A252170 Smallest primitive prime factor of 12^n-1.

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