A007615 -id:A007615 - cp's OEIS frontend

A007498 Unique period lengths of primes mentioned in A007615.

1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Let {Zs(m, 10, 1)} be the Zsigmondy numbers for a = 10, b = 1: Zs(m, 10, 1) is the greatest divisor of 10^m - 1^m that is coprime to 10^r - 1^r for all positive integers r < m. Then this sequence gives m such that Zs(m, 10, 1) is a prime power (e.g., Zs(1, 10, 1) = 9 = 3^2, Zs(2, 10, 1) = 11, Zs(3, 10, 1) = 37, Zs(4, 10, 1) = 101). It is very likely that Zs(m, 10, 1) is prime if m > 1 is in this sequence (note that the Mathematica and PARI programs below are based on this assumption). - Jianing Song, Aug 12 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Ray Chandler, Table of n, a(n) for n = 1..106
Chris K. Caldwell, Unique (period) primes and the factorization of cyclotomic polynomials minus one, Mathematica Japonica, 26 (1997), 189-195.
Chris K. Caldwell & H. Dubner, Unique-Period Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A002371, A048595, A006883, A007732, A051626, A046108.

Cf. A161508 (unique period lengths in base 2).

lst={1}; Do[p=Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], AppendTo[lst, n]], {n, 3000}]; lst (* T. D. Noe, Sep 08 2005 *)

isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ Michel Marcus, Jun 20 2018

More terms from T. D. Noe, Sep 08 2005

a(48)-a(52) from Ray Chandler, Jul 09 2008

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 90, 3, 769230, 714285, 6, 0, 5882352941176470, 5, 526315789473684210, 0, 476190, 45, 4347826086956521739130, 6, 0, 384615, 370, 571428, 3448275862068965517241379310, 3, 322580645161290, 0, 30, 2941176470588235, 285714, 7

Offset: 1

Floor van Lamoen

a(n) = 0 iff n = 2^i*5^j (A003592). - Jon Perry, Nov 19 2014

a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020

			1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.

Philippe Guglielmetti, Table of n, a(n) for n = 1..1001 (first 500 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
See also A060282, A060283, A060251.
A051628 is length of preamble.

isCycl := proc(n) local ifa,i ; if n <= 2 then RETURN(false) ; fi ; ifa := ifactors(n)[2] ; for i from 1 to nops(ifa) do if op(1,op(i,ifa)) <> 2 and op(1,op(i,ifa)) <> 5 then RETURN(true) ; fi ; od ; RETURN(false) ; end: A036275 := proc(n) local ifa,sh,lpow,mpow,r ; if not isCycl(n) then RETURN(0) ; else lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then r := (10^lpow-10^mpow)/n ; r := r mod (10^(lpow-mpow)-1) ; while r*10 < 10^(lpow-mpow) do r := 10*r ; od ; RETURN(r) ; fi ; od ; lpow := lpow+1 ; od ; fi ; end: for n from 1 to 60 do printf("%d %d ",n,A036275(n)) ; od ; # R. J. Mathar, Oct 19 2006

fc[n_]:=Block[{q=RealDigits[1/n][[1,-1]]},If[IntegerQ[q],0,While[First[q]==0,q=RotateLeft[q]];FromDigits[q]]];
Table[fc[n],{n,36}] (* Ray Chandler, Nov 19 2014, corrected Jun 27 2017 *)
Table[FromDigits[FindTransientRepeat[RealDigits[1/n,10,120][[1]],3] [[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2019 *)

Corrected and extended by N. J. A. Sloane

Corrected a(92), a(208), a(248), a(328), a(352) and a(488) which missed a trailing zero (see the table). - Philippe Guglielmetti, Jun 20 2017

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Original entry on oeis.org

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091

Offset: 1

Jud McCranie

Prime p=3 is the only known example of a unique period prime such that A019328(r)/gcd(A019328(r),r) = p^k with k > 1 (cf. A323748). It is plausible to assume that no other such prime exists. Under this (unproved) assumption, the current sequence lists all unique period primes in order and represents a sorted version of A007615. - Max Alekseyev, Oct 14 2022

			The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 1..47
Chris Caldwell, The Prime Glossary, Unique prime.
C. K. Caldwell, "Top Twenty" page, Unique.
Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Unique Prime.
Wikipedia, Unique prime.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.

lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* Arkadiusz Wesolowski, May 13 2012 *)

For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - Max Alekseyev, Oct 14 2022

Missing term a(45) inserted in b-file at the suggestion of Eric Chen by Max Alekseyev, Oct 13 2022

Edited by Max Alekseyev, Oct 14 2022

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

A161509 The unique primitive prime factor of 2^n-1 for the n in A161508.

Original entry on oeis.org

3, 7, 5, 31, 127, 17, 73, 11, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 241, 2731, 262657, 331, 2147483647, 65537, 599479, 43691, 174763, 61681, 5419, 2796203, 4432676798593, 87211, 15790321, 2305843009213693951, 715827883

Offset: 1

T. D. Noe, Jun 17 2009

nonn

For these primes p, the binary expansion of 1/p has a unique period length. The binary analog of A007615.

Max Alekseyev, Table of n, a(n) for n = 1..179 (terms for n=1..100 from T. D. Noe)

Cf. A144755 (sorted).

Reap[Do[c=Cyclotomic[n,2]; q=c/GCD[c,n]; If[PrimePowerQ[q], Sow[FactorInteger[q][[1,1]]]],{n,100}]][[2,1]]

A204847 Primitive cofactor of n-th repunit A002275(n).

Original entry on oeis.org

1, 11, 111, 101, 11111, 91, 1111111, 10001, 333667, 9091, 11111111111, 9901, 1111111111111, 909091, 90090991, 100000001, 11111111111111111, 999001, 1111111111111111111, 99009901, 900900990991, 826446281, 11111111111111111111111, 99990001, 100001000010000100001

Offset: 1

N. J. A. Sloane, Jan 19 2012

nonn

Except for a(1) = 1 and a(3) = 111, this is the Zsigmondy numbers for a = 10, b = 1: Zs(n, 10, 1) is the greatest divisor of 10^n - 1^n that is coprime to 10^m - 1^m for all positive integers m < n. The prime terms are called unique primes or unique period primes (A007615).

Differs from A019328 for n = 1, 9, 22, 27, 42, ... - Jianing Song, Apr 30 2018

Makoto Kamada, Factorizations of 11...11 (Repunit).
Samuel Yates, Cofactors of repunits, Journal of Recreational Mathematics, Vol. 8(2), pp. 99, 1975-76.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A002275, A019328, A102380, A204845, A204846.

lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), f = factor((10^n-1)/9)[,1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k]));); print1(prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf);););} \\ Michel Marcus, May 18 2018

Equals A002275(n)/(product of terms in n-th row of A204845).

a(11)-a(24) from Jianing Song, Apr 30 2018

a(25) from Jinyuan Wang, May 02 2021

A247071 Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.

Original entry on oeis.org

2, 4, 3, 10, 12, 8, 18, 5, 20, 14, 9, 7, 15, 24, 16, 30, 21, 22, 26, 42, 13, 34, 40, 32, 54, 17, 38, 27, 19, 33, 46, 56, 90, 78, 62, 31, 80

Offset: 1

Eric Chen, Nov 16 2014

Periods associated with A144755 in base 2. The binary analog of A051627.

			2^12 - 1 = 4095 = 3 * 3 * 5 * 7 * 13, but none of 3, 5, 7 is a primitive prime factor, so the only primitive prime factor of 2^12 - 1 is 13.

Cf. A161508, A161509, A144755, A007498, A007615, A051627, A040017.

nmax = 65536; primesPeriods = Reap[Do[p = Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]]

a(n) = A002326((A144755(n+1)-1)/2). - Max Alekseyev, Feb 11 2024

Sequence trimmed to the established terms of A144755 by Max Alekseyev, Feb 11 2024

cp's OEIS Frontend

A007498 Unique period lengths of primes mentioned in A007615.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Extensions

A161509 The unique primitive prime factor of 2^n-1 for the n in A161508.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A204847 Primitive cofactor of n-th repunit A002275(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A247071 Numbers n such that 2^n-1 has only one primitive prime factor, sorted according to the magnitude of the corresponding prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula