A019328 -id:A019328 - cp's OEIS frontend

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

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

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

A085035 Number of prime factors of cyclotomic(n,10), which is A019328(n), the value of the n-th cyclotomic polynomial evaluated at x=10.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057951, number of prime factors of 10^n-1.

See references at A085021.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of Phi_n(10)

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), this sequence (x=10).

Cf. A001222, A003060, A005422, A007138, A019328, A057951, A070528, A059892.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 10]]][[2]], {n, 1, 100}]

a(n) = A001222(A019328(n)). - Ray Chandler, May 10 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

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

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

A147556 Largest prime factor of prime(n)-th repunit number.

Original entry on oeis.org

11, 37, 271, 4649, 513239, 265371653, 5363222357, 1111111111111111111, 11111111111111111111111, 77843839397, 57336415063790604359, 2212394296770203368013, 201763709900322803748657942361

Offset: 1

Farideh Firoozbakht, Dec 26 2008

nonn

The sequence of repunit primes is a subsequence of this sequence.

			Prime(15)=47 and (10^47-1)/9 = 35121409*316362908763458525001406154038726382279, so a(15)=316362908763458525001406154038726382279.

Table of n, a(n) for n = 1..70
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A000040, A002275, A003020, A004022, A006530, A019328, A147555.

Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[-1,1]],{n,Prime[ Range[15]]}] (* Harvey P. Dale, Feb 23 2016 *)

a(n) = A003020(A000040(n)) = A006530(A002275(A000040(n))) = A006530(A019328(A000040(n))). - Ray Chandler, May 11 2017

Edited by Ray Chandler, Apr 06 2011
terms to a(66) in b-file from Ray Chandler, May 11 2017
a(67)-a(70) in b-file from Max Alekseyev, Apr 26 2022

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

A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

Original entry on oeis.org

1, 101, 1001001, 1000100010001, 100001000010000100001, 1000001000001000001000001000001, 1000000100000010000001000000100000010000001, 100000001000000010000000100000001000000010000000100000001, 1000000001000000001000000001000000001000000001000000001000000001

Offset: 1

M. F. Hasler, Jan 08 2015

When written in base 10, the terms consist of n digits '1' separated by strings of n-1 digits '0'.
This sequence is relevant for counterexamples to a conjecture in A086766: If p is prime and a(p) is not prime, then A086766(10^(p-1)) = 0.
a(n) is the product of A019328(d) for all d that divide n^2 but not n. - Robert Israel, Jan 08 2015
If a(n) is a prime then n is a prime. What is the smallest prime term greater than 101 in this sequence? - Farideh Firoozbakht, Jan 08 2015
According to what precedes, a(n) is prime iff A019328(d) is prime, where d is the only divisor of n^2 which is not a divisor of n, i.e., iff n is a prime and n^2 is in A138940. No such term is known, except for n=2. - M. F. Hasler, Jan 09 2015

Cf. A128889 (for 2 instead of 10).

seq((10^(n^2)-1)/(10^n-1), n=1..20); # Robert Israel, Jan 08 2015

PARI
```
A252491(n)=(10^(n^2)-1)\(10^n-1)
```

A147555 Smallest prime factor of prime(n)-th repunit number.

Original entry on oeis.org

11, 3, 41, 239, 21649, 53, 2071723, 1111111111111111111, 11111111111111111111111, 3191, 2791, 2028119, 83, 173, 35121409, 107, 2559647034361, 733, 493121, 241573142393627673576957439049, 12171337159

Offset: 1

Farideh Firoozbakht, Dec 26 2008

nonn

The sequence of repunit primes is a subsequence of this sequence.

			prime(15)=47 and (10^47-1)/9 = 35121409*316362908763458525001406154038726382279, so a(15)=35121409.

Ray Chandler, Table of n, a(n) for n = 1..96
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A000040, A002275, A004022, A019328, A020639, A067063, A147556.

a(n) = A067063(A000040(n)) = A020639(A002275(A000040(n))) = A020639(A019328(A000040(n))). - Ray Chandler, May 11 2017

Edited by Ray Chandler, Apr 06 2011

cp's OEIS Frontend

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

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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).

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A085035 Number of prime factors of cyclotomic(n,10), which is A019328(n), the value of the n-th cyclotomic polynomial evaluated at x=10.

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

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

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A147556 Largest prime factor of prime(n)-th repunit number.

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