A007138 -id:A007138 - cp's OEIS frontend

A003060 Smallest number with reciprocal of period length n in decimal (base 10).

1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, 21649, 707, 53, 2629, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 511, 21401, 583, 243, 29, 3191, 211, 2791, 353, 67, 103, 71, 1919, 2028119, 909090909090909091

Offset: 0

N. J. A. Sloane

For n > 0, a(n) is the least divisor d > 1 of 10^n - 1 such that the multiplicative order of 10 mod d is n. For prime n > 3, a(n) = A007138(n). - T. D. Noe, Aug 07 2007

For n > 1, a(n) is the smallest positive d such that d divides 10^n - 1 and does not divide any of 10^k - 1 for 0 < k < n. - Maciej Ireneusz Wilczynski, Sep 06 2012, corrected by M. F. Hasler, Jun 28 2022. (For n = 1, d = 1 divides 10^n - 1 and does not divide any 10^k - 1 with 0 < k < n, but a(1) = 3 > 1.)

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
"Cycle lengths of reciprocals", Popular Computing (Calabasas, CA), Vol. 1 (No. 4, Jul 1973), pp. 12-14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Smallest primitive divisors of b^n-1: A212953 (b=2), A218356 (b=3), A218357 (b=5), A218358 (b=7), this sequence (b=10), A218359 (b=11), A218360 (b=13), A218361 (b=17), A218362 (b=19), A218363 (b=23), A218364 (b=29).

Cf. A007138, A057951, A005422, A176973.

a[n_] := First[ Select[ Divisors[10^n - 1], MultiplicativeOrder[10, #] == n &, 1]]; a[0] = 1; a[1] = 3; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 13 2012, after T. D. Noe *)

apply( {A003060(n)=!fordiv(10^n-!!n, d, d>1 && znorder(Mod(10,d))==n && return(d))}, [0..50]) \\ M. F. Hasler, Jun 28 2022

Comment corrected by T. D. Noe, Apr 15 2010
More terms from T. D. Noe, Apr 15 2010
b-file truncated at uncertain term a(439) by Max Alekseyev, Apr 30 2022

A072982 Primes p for which the period of 1/p is a power of 2.

Original entry on oeis.org

3, 11, 17, 73, 101, 137, 257, 353, 449, 641, 1409, 10753, 15361, 19841, 65537, 69857, 453377, 976193, 1514497, 5767169, 5882353, 6187457, 8253953, 8257537, 70254593, 167772161, 175636481, 302078977, 458924033, 639631361, 1265011073

Offset: 1

Benoit Cloitre, Jul 26 2002

All Fermat primes > 5 (A019434) are in the sequence, since it can be shown that the period of 1/(2^(2^n)+1) is 2^(2^n) whenever 2^(2^n)+1 is prime. - Benoit Cloitre, Jun 13 2007
Take all the terms from row 2^k of triangle in A046107 for k >= 0 and sort to arrive at this sequence. - Ray Chandler, Nov 04 2011
Additional terms, but not necessarily the next in sequence: 13462517317633 has period 1048576 = 2^20; 46179488366593 has period 2199023255552 = 2^41; 101702694862849 has period 8388608 = 2^23; 171523813933057 has period 4398046511104 = 2^42; 505775348776961 has period 2199023255552 = 2^41; 834427406578561 has period 64 = 2^6 - Ray Chandler, Nov 09 2011
Furthermore (excluding the initial term 3) this sequence is also the ascending sequence of primes dividing 10^(2^k)+1 for some nonnegative integer k. For a prime dividing 10^(2^k)+1, the period of 1/p is 2^(k+1). Thus for the prime p = 558711876337536212257947750090161313464308422534640474631571587847325442162307811\
65223702155223678309562822667655169, a factor of 10^(2^7)+1, the period of 1/p is only 2^8. This large prime then also belongs to the sequence. - Christopher J. Smyth, Mar 13 2014
For any m, every term that is not a factor of 10^(2^k)-1 for some k < m is congruent to 1 (mod 2^m). Thus all terms except 3, 11, 17, 73, 101, 137, 353, 449, 69857, 976193, 5882353, 6187457 are congruent to 1 (mod 128). - Robert Israel, Jun 17 2016
Additional terms listed earlier confirmed as next terms in sequence. - Arkadiusz Wesolowski, Jun 17 2016

			15361 has a period of 256 = 2^8, hence 15361 is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..45 (first 33 terms from Ray Chandler, to 36 terms from Robert G. Wilson v, to 39 terms from Ray Chandler)
Ray Chandler, Known Terms of A072982
Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status
Index entries for sequences related to decimal expansion of 1/n

Cf. A002371, A007138, A046107, A054471.

Cf. A197224 (power of 2 which is the period of the decimal 1/a(n)).

filter:= proc(p) local k;
  if not isprime(p) then return false fi;
  k:=igcd(p-1,2^ilog2(p));
  evalb(10 &^ k mod p = 1)
end proc:
r:= select(`<=`,`union`(seq(numtheory:-factorset(10^(2^k)-1),k=1..6)),10^9):
b:= select(filter, {seq(i,i=129..10^9,128)}):
sort(convert(r union b, list)); # Robert Israel, Jun 17 2016

Do[ If[ IntegerQ[ Log[2, Length[ RealDigits[ 1/Prime[n]] [[1, 1]]]]], Print[ Prime[n]]], {n, 1, 47500}] (* Robert G. Wilson v, May 09 2007 *)
pmax = 10^10; p = 1; While[p < pmax,p = NextPrime[p];If[ IntegerQ[Log[2, MultiplicativeOrder[10, p] ] ], Print[ p];];]; (* Ray Chandler, May 14 2007 *)

select( {is_A072982(p)=if(p>5, 1<M. F. Hasler, Nov 18 2024

from itertools import count, islice
from sympy import prime, n_order
def A072982_gen(): return (p for p in (prime(n) for n in count(2)) if p != 5 and bin(n_order(10,p))[2:].rstrip('0') == '1')
A072982_list = list(islice(A072982_gen(),10)) # Chai Wah Wu, Feb 07 2022

from sympy import primerange, n_order
A072982_upto = lambda N=1e5: [p for p in primerange(3, N) if p != 5 and n_order(10, p).bit_count() == 1] # or (...) to get a generator. - M. F. Hasler, Nov 19 2024

Edited by Robert G. Wilson v, Aug 20 2002
a(18) from Ray Chandler, May 02 2007
a(19) from Robert G. Wilson v, May 09 2007
a(20)-a(32) from Ray Chandler, May 14 2007
Deleted an unsatisfactory PARI program. - N. J. A. Sloane, Nov 19 2024

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)), ", "); ); }

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)), ", "); ); }

A066364 Prime divisors of solutions to 10^n == 1 (mod n).

Original entry on oeis.org

3, 37, 163, 757, 1999, 5477, 8803, 9397, 13627, 15649, 36187, 40879, 62597, 106277, 147853, 161839, 215893, 231643, 281683, 295759, 313471, 333667, 338293, 478243, 490573, 607837, 647357, 743933, 988643, 1014877, 1056241, 1168711, 1353173, 1390757, 1487867, 1519591, 1627523, 1835083, 1912969, 2028119, 2029759, 2064529

Offset: 1

Vladeta Jovovic, Dec 21 2001

nonn

			10^27-1 = 3^5*37*757*333667*440334654777631 is a solution to the congruence.

Max Alekseyev and Hans Havermann (Max Alekseyev to 501), Table of n, a(n) for n = 1..2060
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A014950, A001270, A027889, A007138, A114207.

fQ[p_] := Block[{fi = First@# & /@ FactorInteger[ MultiplicativeOrder[ 10, p]]}, Union[ MemberQ[ lst, #] & /@ fi] == {True}]; k = 4; lst = {3}; While[k < 180000, If[ p = Prime@ k; fQ@ p, AppendTo[ lst, p]; Print@ p]; k++]; lst (* Robert G. Wilson v, Nov 30 2013 *)

S=Set([3]); forprime(p=7,10^6, v=factorint(znorder(Mod(10,p)))[,1]; if(length(setintersect(S,Set(v)))==length(v), S=setunion(S,[p])) ); print(vecsort(eval(S))) \\ Max Alekseyev, Nov 16 2005

A prime p is a term iff all prime divisors of ord_p(10) are terms, where ord_p(10) is the order of 10 modulo p. - Max Alekseyev, Nov 16 2005

Edited by Max Alekseyev, Nov 16 2005

Edited by Hans Havermann, Jul 11 2014

A061075 Greatest prime number p(n) with decimal fraction period of length n.

Original entry on oeis.org

3, 11, 37, 101, 271, 13, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 52579, 1111111111111111111, 27961, 10838689, 8779, 11111111111111111111111, 99990001, 182521213001, 1058313049, 440334654777631, 121499449, 77843839397

Offset: 1

Heiner Muller-Merbach (hmm(AT)sozwi.uni-kl.de), May 29 2001

			1/271 = 0.0036900369, period of n=5 for p(5)=271.

Cf. A003020, A005422, A006530, A007138, A019328.

Last terms in rows of A046107.

a[n_] := Cyclotomic[n, 10] // FactorInteger // Last // First; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Aug 05 2013, after Pari *)

a(n) = my(p); if(n<1, 0, p=factor(polcyclo(n,10))[,1]; p[#p])

a(n) = A006530(A019328(n)). - Ray Chandler, May 10 2017

Terms to a(322) in b-file from Ray Chandler, Apr 28 2017

a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

A112505 Number of primitive prime factors of 10^n-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 3, 3, 1, 1, 3, 2, 2, 3, 5, 3, 3, 5, 2, 3, 3, 1, 3, 1, 1, 2, 4, 3, 4, 3, 2, 4, 2, 1, 2, 3, 4, 2, 4, 2, 4, 2, 3, 2, 2, 3, 7, 1, 5, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 4, 4, 6, 2, 5, 2, 3, 2, 3, 3, 3, 2, 5, 3, 7, 3, 1, 3, 5, 4, 3, 2, 4, 4

Offset: 1

T. D. Noe, Sep 08 2005

Also the number of primes whose reciprocal is a repeating decimal of length n. The number of numbers in each row of table A046107.

By Zsigmondy's theorem, a(n) >= 1. When a(n)=1, the corresponding prime is called a unique prime (see A007498, A040017 and A051627).

Table of n, a(n) for n = 1..352
Eric Weisstein's World of Mathematics, Primitive Prime Factor
Eric Weisstein's World of Mathematics, Zsigmondy Theorem
Eric Weisstein's World of Mathematics, Unique Prime

Cf. A007138 (smallest primitive prime factor of 10^n-1), A102347 (number of distinct prime factors of 10^n-1), A046107.

pp={}; Table[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Length[p], {n, 66}]

Terms to a(276) in b-file from T. D. Noe, Jun 01 2010
a(277)-a(322) in b-file from Ray Chandler, May 01 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 28 2022

cp's OEIS Frontend

A003060 Smallest number with reciprocal of period length n in decimal (base 10).

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A072982 Primes p for which the period of 1/p is a power of 2.

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A379640 Smallest primitive prime factor of 7^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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A379639 Smallest primitive prime factor of 6^n-1.

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

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A066364 Prime divisors of solutions to 10^n == 1 (mod n).

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