A046107 -id:A046107 - cp's OEIS frontend

A046108 Decimal period of 1/b(n), where b(n) is A046107.

1, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35

Offset: 1

Eric W. Weisstein

nonn

Max Alekseyev, Table of n, a(n) for n = 1..1248
Eric Weisstein's World of Mathematics, Decimal Expansion.

Cf. A046107, A007498.

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

A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of 11...11 (Repunit).
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): this sequence (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.

PrimeOmega[10^Range[70]-1] (* Jayanta Basu, May 29 2013 *)

a(n)=bigomega(10^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085035 - T. D. Noe, Jun 19 2003
a(n) = Omega(10^n -1) = Omega(R_n) + 2 = A046053(n) + 2 {where Omega(n) = A001222(n) and R_n = (10^n - 1)/9 = A002275(n)}. - Lekraj Beedassy, Jun 09 2006
a(n) = A001222(A002283(n)). - Ray Chandler, Apr 22 2017

Erroneous b-file replaced by Ray Chandler, Apr 26 2017

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

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

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

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

A106305 Divisors of 10^14 - 1.

Original entry on oeis.org

1, 3, 9, 11, 33, 99, 239, 717, 2151, 2629, 4649, 7887, 13947, 23661, 41841, 51139, 153417, 460251, 909091, 1111111, 2727273, 3333333, 8181819, 9999999, 10000001, 12222221, 30000003, 36666663, 90000009, 109999989, 217272749, 651818247

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 12 2005

Nathaniel Johnston, Table of n, a(n) for n = 1..48 (full sequence)
Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A001270, A007138, A046107, A112505.

Divisors[10^14 - 1] (* Harvey P. Dale, Feb 01 2015 *)

divisors(10^14-1) \\ Charles R Greathouse IV, Jun 21 2017

10^14 - 1 = 3^2 * 11 * 239 * 4649 * 909091 = 99999999999999. - Alonso del Arte, Nov 09 2017

A109933 Divisors of 10^13 - 1.

Original entry on oeis.org

1, 3, 9, 53, 79, 159, 237, 477, 711, 4187, 12561, 37683, 265371653, 796114959, 2388344877, 14064697609, 20964360587, 42194092827, 62893081761, 126582278481, 188679245283, 1111111111111, 3333333333333, 9999999999999

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 11 2005

Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A001270, A007138, A046107.

Divisors[10^13 - 1] (* Wesley Ivan Hurt, Feb 14 2014 *)

divisors(10^13-1) \\ Charles R Greathouse IV, Jun 21 2017

A187614 Primes p such that the decimal representation of 1/p does not contain every digit 0-9.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 31, 37, 41, 43, 67, 73, 79, 101, 137, 239, 271, 353, 449, 757, 859, 1933, 4649, 8779, 9091, 9901, 21401, 21649, 25601, 27961, 52579, 62003, 123551, 333667, 513239, 538987, 909091, 1676321, 2071723, 2906161, 5882353, 10838689, 35121409, 52986961, 99990001, 265371653, 1056689261, 1058313049, 1360682471

Offset: 1

Michel Lagneau, Mar 12 2011

Every repunit prime (A004022) is here. There are 113 terms of A046107, having periods of up to 256, that are here. The only known unique-period prime (A007615) not here is the one having period 92092. Is this sequence finite? - T. D. Noe, Mar 13 2011

			4649 is in the sequence because 1/4649 = 0.00021510002151000215.... contain
  only the digits 0, 1, 2 and 5.

Cf. A187372.

Cf. A352023 (does not contain digit 9)

Join[{2, 3, 5}, Select[Prime[Range[4, 10000]], Length[Union[RealDigits[1/#][[1, 1]]]] < 10 &]]

from sympy import n_order, nextprime
from itertools import islice
def A187614_gen(): # generator of terms
    yield from (2,3,5)
    p = 7
    while True:
        if len(set('0'+str(10**(n_order(10, p))//p))) < 10:
            yield p
        p = nextprime(p)
A187614_list = list(islice(A187614_gen(),20)) # Chai Wah Wu, Mar 03 2022

Extended by T. D. Noe, Mar 12 2011

cp's OEIS Frontend

A046108 Decimal period of 1/b(n), where b(n) is A046107.

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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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A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

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

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

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A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

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

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A106305 Divisors of 10^14 - 1.

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