A379640 -id:A379640 - cp's OEIS frontend

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

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

A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211

Offset: 1

Vladimir Shevelev, Aug 25 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);
a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.
If a(n) > 1 then a(n) is the index where n occurs first in A014664. - M. F. Hasler, Feb 21 2016
Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - Jeppe Stig Nielsen, Aug 31 2020

Robert G. Wilson v, Table of n, a(n) for n = 1..1206
Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Will Edgington, Factored Mersenne Numbers [from Internet Archive Wayback Machine]
Wikipedia, Zsigmondy's theorem

Cf. A002326, A064078, A001122, A115591.

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).

A112927(n,f=factor(2^n-1)[,1])=!for(i=1,#f,znorder(Mod(2,f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1,0)[,1]. - M. F. Hasler, Feb 21 2016

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

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)), ", "); ); }

A379642 Smallest primitive prime factor of 9^n-1.

Original entry on oeis.org

2, 5, 7, 41, 11, 73, 547, 17, 19, 1181, 23, 6481, 398581, 29, 31, 21523361, 103, 530713, 1597, 42521761, 43, 5501, 47, 97, 151, 53, 109, 430697, 59, 47763361, 683, 926510094425921, 25411, 956353, 71, 282429005041, 18427, 5301533, 79, 14401, 83, 2857, 431, 89

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..730 (first 690 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. A274909.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(9^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A379639 Smallest primitive prime factor of 6^n-1.

Original entry on oeis.org

5, 7, 43, 37, 311, 31, 55987, 1297, 19, 11, 23, 13, 3433, 29, 1171, 17, 239, 46441, 191, 241, 1822428931, 51828151, 47, 1678321, 18198701, 53, 163, 421, 7369130657357778596659, 1950271, 5333, 353, 67, 190537, 71, 73, 149, 1787, 3143401, 41, 8648131, 2527867231

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..436 (first 420 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. A274907.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(6^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

A379644 Smallest primitive prime factor of 11^n-1.

Original entry on oeis.org

2, 3, 7, 61, 3221, 37, 43, 7321, 1772893, 13421, 15797, 13, 1093, 1623931, 195019441, 17, 50544702849929377, 590077, 6115909044841454629, 212601841, 1723, 23, 829, 10657, 3001, 53, 5559917315850179173, 29, 523, 31, 50159, 51329, 661, 71707, 211, 3138426605161

Offset: 1

Sean A. Irvine, Dec 28 2024

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..330 (first 253 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. A274910.

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(11^n-1)[, 1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "); ); }

cp's OEIS Frontend

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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A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

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

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A379642 Smallest primitive prime factor of 9^n-1.

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