A040017 -id:A040017 - cp's OEIS frontend

A051627 Periods associated with A040017.

1, 2, 3, 4, 10, 12, 9, 14, 24, 36, 48, 38, 19, 23, 39, 62, 120, 150, 106, 93, 134, 294, 196, 320, 654, 738, 385, 586, 317, 597, 1404, 945, 1452, 1836, 1752, 1172, 1812, 1282, 1426, 2232, 1862, 1844, 1521, 2134, 3750, 1031, 2264, 2667, 4354, 3927, 4274, 6522, 3903, 6022, 6682, 6135, 9550, 5877

Offset: 1

N. J. A. Sloane

The numbers in A007498 sorted according to the magnitude of the corresponding prime. - T. D. Noe, Sep 08 2005

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

Ray Chandler, Table of n, a(n) for n = 1..99
C. K. Caldwell, Unique Primes
Eric Weisstein's World of Mathematics, Unique Prime
Index entries for sequences related to decimal expansion of 1/n

nmax = 10000; primesPeriods = Reap[Do[p = Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], Print[n]; Sow[{p, n}]], {n, 1, nmax}]][[2, 1]]; Sort[primesPeriods][[All, 2]] // Prepend[#, 1]& // Take[#, 58]& (* Jean-François Alcover, Mar 29 2013 *)

a(n) = A002371(A000720(A040017(n))). - Max Alekseyev, Oct 14 2022

More terms from Jud McCranie
More terms from T. D. Noe, Sep 08 2005
Corrected a(45)=3750 and extended by Ray Chandler, Oct 13 2008

A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

2, 4, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 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, 5877, 6022

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Unique period primes (A040017) are often of the form Phi(k,10) or Phi(k,-10).

Terms of this sequence which are the square of a prime, a(n)=p^2, are such that A252491(p) is prime. Apart from a(2)=2^2, there is no such term up to 26570. - M. F. Hasler, Jan 09 2015

Ray Chandler, Table of n, a(n) for n = 1..102 (first 50 terms from Robert Price, terms 92-93 from Serge Batalov, others from Kamada link)
Chris Caldwell, Unique Primes.
Makoto Kamada, Factorizations of Phi_n(10) (including prime members up to 200000).
Index entries for cyclotomic polynomials, values at X

Cf. A019328, A040017, A085035, A252491.

Cf. Subsequence of A007498, contains A004023.

Select[Range[1000], PrimeQ[Cyclotomic[#, 10]] &] (* T. D. Noe, Mar 03 2012 *)

for( i=1,999, isprime( polcyclo(i,10)) && print1( i","))

a(28)-a(43) from Robert Price, Mar 03 2012
a(44)-a(50) from Robert Price, Apr 14 2012
a(51)-a(91) from Ray Chandler, Maksym Voznyy et al. (cf. Phi_n(10) link), ca. 2009
a(92)-a(93) from Serge Batalov, Mar 28 2015

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

A138920 Indices k such that A020509(k)=Phi[k](-10) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

4, 5, 7, 12, 19, 24, 31, 36, 38, 46, 48, 53, 67, 75, 78, 120, 186, 196, 293, 320, 327, 369, 634, 641, 713, 770, 931, 1067, 1172, 1194, 1404, 1452, 1752, 1812, 1836, 1844, 1875, 1890, 2062, 2137, 2177, 2232, 2264, 3011, 3042, 3261, 3341, 4775, 5334, 6685

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

"Unique [period] primes" (A040017) are often of the form Phi[k](10) or Phi[k](-10).

Two cyclotomic polynomial identities tightly connect this sequence to A138940: 1) Phi_2k(x) = Phi_k(-x) for odd integer k > 1. 2) Phi_4k(x) = Phi_2k(x^2) for all positive integer k. - Ray Chandler, Apr 30 2017

Ray Chandler, Table of n, a(n) for n = 1..89 (first 76 terms from Robert Price)
C. Caldwell, Unique Primes.
Index entries for cyclotomic polynomials, values at X

Cf. A019328, A040017, A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, -10]] &]

for( i=1,999, is/*pseudo*/prime( polcyclo(i,-10)) &&& print1( i",")) /* for PARI < 2.4.2 use ...subst(polcyclo(i,x),x,-10)... */

a(28)-a(43) from Robert Price, Mar 09 2012

a(44)-a(50) from Robert Price, Apr 14 2012

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

A144755 Primes which divide none of overpseudoprimes to base 2 (A141232).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, 2731, 5419, 8191, 43691, 61681, 65537, 87211, 131071, 174763, 262657, 524287, 599479, 2796203, 15790321, 18837001, 22366891, 715827883, 2147483647, 4278255361

Offset: 1

Vladimir Shevelev, Sep 20 2008

nonn

Odd prime p is in the sequence iff A064078(A002326((p-1)/2))=p. For example, for p=127 we have A002326((127-1)/2)=7 and A064078(7)=127. Thus p=127 is in the sequence.
Primes p such that the binary expansion of 1/p has a unique period length; that is, no other prime has the same period. Sequence A161509 sorted. - T. D. Noe, Apr 13 2010
Since A161509 has terms of varying magnitude, sorting any finite initial segment of A161509 cannot provide a guarantee that there are no other terms missed in between. Any prime p not (yet) appearing in A161509 should be tested via A064078(A002326((p-1)/2))=p to conclude whether it belongs to the current sequence. - Max Alekseyev, Feb 10 2024

			Overpseudoprimes to base 2 are odd, then a(1)=2.

Cf. A141232, A002326, A064078, A112927.

Cf. A040017 (unique-period primes in base 10). - T. D. Noe, Apr 13 2010

b=2; t={}; Do[c=Cyclotomic[n,b]; q=c/GCD[n,c]; If[PrimePowerQ[q], p=FactorInteger[q][[1,1]]; If[p<10^12, AppendTo[t,p]; Print[{n,p}]]], {n,1000}]; t=Sort[t] (* T. D. Noe, Apr 13 2010 *)

{ is_a144755(p) = my(q,m,g); q=znorder(Mod(2,p)); m=2^q-1; fordiv(q,d, if(d1,m\=g))); m==p; } \\ Max Alekseyev, Feb 10 2024

Extended by T. D. Noe, Apr 13 2010

b-file deleted by Max Alekseyev, Feb 10 2024.

A007615 Primes with unique period length (the periods are given in A007498).

Original entry on oeis.org

3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991

Offset: 1

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

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

			3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

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.

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..25 from T. D. Noe; terms 26..31 from Ray Chandler)
C. K. Caldwell, The Prime Glossary, unique prime
Makoto Kamada, Factorizations of Phi_n(10)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)

a(n) = A061075(A007498(n)). - Max Alekseyev, Oct 16 2010

a(n) = A006530(A019328(A007498(n))). - Ray Chandler, May 10 2017

A306073 Bases in which 3 is a unique-period prime.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 17, 23, 26, 28, 35, 47, 53, 71, 80, 82, 95, 107, 143, 161, 191, 215, 242, 244, 287, 323, 383, 431, 485, 575, 647, 728, 730, 767, 863, 971, 1151, 1295, 1457, 1535, 1727, 1943, 2186, 2188, 2303, 2591, 2915, 3071, 3455, 3887

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 = 3^t + 1, t >= 1; (b) b = 2^s*3^t - 1, s >= 0, t >= 1.
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 = 3, there are no nontrivial bases, since ord(3,b) <= 2.

			If b = 3^t + 1, t >= 1, then b - 1 only has prime factor 3, so 3 is a unique-period prime in base b.
If b = 2^s*3^t - 1, t >= 1, then the prime factors of b^2 - 1 are 3 and prime factors of b - 1 = 2^s*3^t - 2, 3 is the only new prime factor so 3 is a unique-period prime in base b.

Jianing Song, Table of n, a(n) for n = 1..805
Wikipedia, Unique prime

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

Bases in which p is a unique prime: A000051 (p=2), this sequence (p=3), A306074 (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

p = 3;
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, ", "))));

A306074 Bases in which 5 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 19, 24, 26, 39, 49, 79, 99, 124, 126, 159, 199, 249, 319, 399, 499, 624, 626, 639, 799, 999, 1249, 1279, 1599, 1999, 2499, 2559, 3124, 3126, 3199, 3999, 4999, 5119, 6249, 6399, 7999, 9999, 10239, 12499, 12799, 15624, 15626, 15999, 19999, 20479

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 = 5^t + 1, t >= 1; (b) b = 2^s*5^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 7.
For every odd prime p, p is a 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 = 5, the nontrivial bases are 2, 3, 7.

			1/5 has period length 4 in base 2. Note that 3 and 5 are the only prime factors of 2^4 - 1 = 15, but 1/3 has period length 2, so 5 is a unique-period prime in base 2.
1/5 has period length 4 in base 3. Note that 2 and 5 are the only prime factors of 3^4 - 1 = 80, but 1/2 has period length 1, so 5 is a unique-period prime in base 3.
1/5 has period length 4 in base 7. Note that 2, 3 and 5 are the only prime factors of 7^4 - 1 = 2400, but 1/2 and 1/3 both have period length 1, so 5 is a unique-period prime in base 7.

Jianing Song, Table of n, a(n) for n = 1..544
Wikipedia, Unique prime

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

Bases in which p is a unique-period prime: A000051 (p=2), A306073 (p=3), this sequence (p=5), A306075 (p=7), A306076 (p=11), A306077 (p=13).

p = 5;
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, ", "))));

A306075 Bases in which 7 is a unique-period prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 13, 18, 19, 27, 48, 50, 55, 97, 111, 195, 223, 342, 344, 391, 447, 685, 783, 895, 1371, 1567, 1791, 2400, 2402, 2743, 3135, 3583, 4801, 5487, 6271, 7167, 9603, 10975, 12543, 14335, 16806, 16808, 19207, 21951, 25087, 28671, 33613, 38415, 43903, 50175

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 = 7^t + 1, t >= 1; (b) b = 2^s*7^t - 1, s >= 0, t >= 1; (c) b = 2, 3, 4, 5, 18, 19.
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 = 7, the nontrivial bases are 2, 3, 4, 5, 18, 19.

			1/7 has period length 3 in base 2. Note that 7 is the only prime factor of 2^3 - 1 = 7, so 7 is a unique-period prime in base 2.
1/7 has period length 3 in base 4. Note that 3, 7 are the only prime factors of 4^3 - 1 = 63, but 1/3 has period length 1, so 7 is a unique-period prime in base 4.
1/7 has period length 3 in base 18. Note that 7, 17 are the only prime factors of 18^3 - 1 = 5831, but 1/17 has period length 1, so 7 is a unique-period prime in base 18.
(1/7 has period length 6 in base 3, 5, 19. Similar demonstrations can be found.)

Jianing Song, Table of n, a(n) for n = 1..448
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), this sequence (p=7), A306076 (p=11), A306077 (p=13).

p = 7;
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, ", "))));

cp's OEIS Frontend

A051627 Periods associated with A040017.

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A138940 Indices n such that A019328(n) = Phi(n,10) is prime, where Phi is a cyclotomic polynomial.

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

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A138920 Indices k such that A020509(k)=Phi[k](-10) is prime, where Phi is a cyclotomic polynomial.

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

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A144755 Primes which divide none of overpseudoprimes to base 2 (A141232).

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A007615 Primes with unique period length (the periods are given in A007498).

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