A040017 -id:A040017 - cp's OEIS frontend

A306076 Bases in which 11 is a unique-period prime.

2, 3, 10, 12, 21, 43, 87, 120, 122, 175, 241, 351, 483, 703, 967, 1330, 1332, 1407, 1935, 2661, 2815, 3871, 5323, 5631, 7743, 10647, 11263, 14640, 14642, 15487, 21295, 22527, 29281, 30975, 42591, 45055, 58563, 61951, 85183, 90111, 117127, 123903, 161050, 161052, 170367, 180223, 234255, 247807, 322101, 340735

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period length of the base-b expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 11^t + 1, t >= 1; (b) b = 2^s*11^t - 1, s >= 0, t >= 1; (c) b = 2, 3.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 11, the nontrivial bases are 2, 3.

			1/11 has period length 10 in base 2. Note that 3, 11, 31 are the only prime factors of 2^10 - 1 = 1023, but 1/3 has period length 2 and 1/31 has period length 5, so 11 is a unique-period prime in base 2.
1/11 has period length 5 in base 3. Note that 2, 11 are the only prime factors of 3^5 - 1 = 242, but 1/2 has period length 1, so 11 is a unique-period prime in base 3.

Jianing Song, Table of n, a(n) for n = 1..590
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), this sequence (p=11), A306077 (p=13).

p = 11;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A306077 Bases in which 13 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 5, 12, 14, 22, 23, 25, 51, 103, 168, 170, 207, 239, 337, 415, 675, 831, 1351, 1663, 2196, 2198, 2703, 3327, 4393, 5407, 6655, 8787, 10815, 13311, 17575, 21631, 26623, 28560, 28562, 35151, 43263, 53247, 57121, 70303, 86527, 106495, 114243, 140607, 173055, 212991, 228487, 281215, 346111

Offset: 1

Jianing Song, Jun 19 2018

A prime p is called a unique-period prime in base b if there is no other prime q such that the period of the base-b expansion of its reciprocal, 1/p, is equal to the period of the reciprocal of q, 1/q.
A prime p is a unique-period prime in base b if and only if Zs(b, 1, ord(b,p)) = p^k, k >= 1. Here Zs(b, 1, d) is the greatest divisor of b^d - 1 that is coprime to b^m - 1 for all positive integers m < d, and ord(b,p) is the multiplicative order of b modulo p.
b is a term if and only if: (a) b = 13^t + 1, t >= 1; (b) b = 2^s*13^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 22, 23, 239.
For every odd prime p, p is a unique-period prime in base b if b = p^t + 1, t >= 1 or b = 2^s*p^t - 1, s >= 0, t >= 1. These are trivial bases in which p is a unique-period prime, with ord(b,p) = 1 or 2. By Faltings's theorem, there are only finitely many nontrivial bases in which p is also a unique-period prime, with ord(b,p) >= 3. For p = 13, the nontrivial bases are 2, 3, 4, 5, 22, 23, 239.

			1/13 has period 12 in base 2. Note that 3, 5, 7, 13, 31 are the only prime factors of 2^12 - 1 = 4095, but 1/3 has period 2, 1/5 has period 4, 1/7 has period 3, 1/31 has period 5, so 13 is a unique-period prime in base 2. (For the same reason, 13 is a unique-period prime in base 4.)
1/13 has period 3 in base 3. Note that 2, 13 are the only prime factors of 3^3 - 1 = 26, but 1/2 has period 1, so 13 is a unique-period prime in base 3.
1/13 has period 3 in base 22. Note that 3, 7, 13 are the only prime factors of 22^3 - 1 = 10647, but 1/3 and 1/7 both have period 1, so 13 is a unique-period prime in base 22.

Jianing Song, Table of n, a(n) for n = 1..554
Wikipedia, Unique prime

Cf. A040017 (unique-period primes in base 10), A144755 (unique-period primes in base 2).

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), this sequence (p=13).

p = 13;
gpf(n)=if(n>1, vecmax(factor(n)[, 1]), 1);
test(n, q)=while(n%p==0, n/=p); if(q>1, while(n%q==0, n/=q)); n==1;
for(n=2, 10^6, if(gcd(n, p)==1, if(test(polcyclo(znorder(Mod(n, p)), n), gpf(znorder(Mod(n, p)))), print1(n, ", "))));

A217611 Primes p such that the octal expansion of 1/p has a unique period length.

Original entry on oeis.org

3, 7, 19, 73, 87211, 262657, 18837001, 77158673929, 5302306226370307681801, 19177458387940268116349766612211, 6113142872404227834840443898241613032969, 328017025014102923449988663752960080886511412965881

Offset: 1

Arkadiusz Wesolowski, Oct 08 2012

Also called generalized unique primes in base 8.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..27
C. K. Caldwell, "Top Twenty" page, Generalized Unique
Wikipedia, Octal

Cf. A019326, A040017 (unique-period primes in base 10).

lst = {}; Do[c = Cyclotomic[n, 8]; q = c/GCD[n, c]; If[PrimePowerQ[q], p = FactorInteger[q][[1, 1]]; AppendTo[lst, p]], {n, 138}]; Sort[lst]

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

A072848 Largest prime factor of 10^(6*n) + 1.

Original entry on oeis.org

9901, 99990001, 999999000001, 9999999900000001, 39526741, 3199044596370769, 4458192223320340849, 75118313082913, 59779577156334533866654838281, 100009999999899989999000000010001, 2361000305507449, 111994624258035614290513943330720125433979169

Offset: 1

Rick L. Shepherd, Jul 25 2002

nonn

According to the link, there are only 18 "unique primes" below 10^50. The first four terms above are each unique primes, of periods 12, 24, 36 and 48, respectively, according to Caldwell and the cross-referenced sequences. These are precisely the only unique primes (less than 10^50 at least) with this type of digit pattern: m 9's, m-1 0's and 1, in that order. (Also a(10) is a unique prime of period 120.)

			10^(6*4)+1 = 17 * 5882353 * 9999999900000001, so a(4) = 9999999900000001, the largest prime factor.

Max Alekseyev, Table of n, a(n) for n = 1..87 (terms n=1..51 from Ray Chandler, n=52..81 from Daniel Suteu)
C. K. Caldwell, Unique Primes
Makoto Kamada, Factorizations of 100...001.

Cf. A040017 (unique period primes), A051627 (associated periods).

Cf. A003021, A006530, A062397.

for(n=1,12,v=factor(10^(6*n)+1); print1(v[matsize(v)[1],1],","))

a(n) = A003021(6n) = A006530(A062397(6n)). - Ray Chandler, May 11 2017

A119761 Values n such that the largest prime factor of repunit R_n=(10^n -1)/9 is a unique prime,i.e.,whose reciprocal has unique decimal period length.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 12, 14, 18, 19, 23, 24, 36, 38, 39, 46, 48, 62, 78, 93, 96, 106, 120, 134, 150, 186, 196, 240, 268, 294, 300, 317, 320

Offset: 1

Lekraj Beedassy, Jun 18 2006

			18 is in the sequence because R_18=3^2*7*11*13*19*37*52579*333667 and 333667 is the only prime with period 9: 1/333667 =0.000002997000002997...

M. Kamada, Factorization of 11...11(Repunits)

Cf. A002275, A003020, A007498, A040017.

Corrected and extended by Hans Havermann, Jun 22 2006

a(25)-a(33) from Ray Chandler, May 09 2017

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A306076 Bases in which 11 is a unique-period prime.

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A306077 Bases in which 13 is a unique-period prime.

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A217611 Primes p such that the octal expansion of 1/p has a unique period length.

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

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A072848 Largest prime factor of 10^(6*n) + 1.

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A119761 Values n such that the largest prime factor of repunit R_n=(10^n -1)/9 is a unique prime,i.e.,whose reciprocal has unique decimal period length.

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