A051200 -id:A051200 - cp's OEIS frontend

A033175 n 3's followed by 1.

1, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333331, 3333333331, 33333333331, 333333333331, 3333333333331, 33333333333331, 333333333333331, 3333333333333331, 33333333333333331, 333333333333333331, 3333333333333333331, 33333333333333333331, 333333333333333333331

Offset: 0

Jan Jensen (dorul(AT)post6.tele.dk)

F. Smarandache, Properties of numbers, University of Craiova, 1973.

Muniru A Asiru, Table of n, a(n) for n = 0..990
Eric Weisstein's World of Mathematics, 3.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A051200.

List([0..20],n->(10^(n+1)-7)/3); # Muniru A Asiru, Oct 31 2018

Table[FromDigits[PadLeft[{1},n,3]],{n,20}] (* Harvey P. Dale, Feb 21 2013 *)

a(n)=(10^(n+1)-7)/3 \\ Charles R Greathouse IV, Oct 07 2015

def a(n): return int('3'*n + '1')
print([a(n) for n in range(20)]) # Michael S. Branicky, Jan 31 2021

a(n) = (10^(n+1) - 7)/3.
a(n) = 10*a(n-1) + 21 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
G.f.: (1+20*x)/((10*x-1)*(x-1)). - R. J. Mathar, Aug 24 2011
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(10*exp(9*x) - 7)/3.
a(n) = 11*a(n-1) - 10*a(n-2). (End)

A055558 Primes of the form 1999...999.

Original entry on oeis.org

19, 199, 1999, 199999, 19999999, 199999999999999999999999999, 1999999999999999999999999999, 199999999999999999999999999999999999999999999999999999

Offset: 1

Labos Elemer, Jul 10 2000

Primes of the form 2*10^k - 1.

			2*10^n - 1 is prime for {1,2,3,5,7,26,27,53,147,236,248,386,401}; in each of these numbers, the digit '9' appears n times.

Subsequence of A090149.
Primes in A067272.
Cf. A002957, A051200.

for(n=1,99,if(ispseudoprime(t=2*10^n-1),print1(t", "))) \\ Charles R Greathouse IV, Dec 10 2013

a(n) = 2*10^A002957(n) - 1 = A067272(A002957(n) + 1). - Elmo R. Oliveira, Jun 14 2025

A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1732, 1918, 8855, 11245, 11960, 12130, 18533, 22718, 23365, 24253, 24549, 25324, 30178, 53718, 380976, 424861, 563535, 666903

Offset: 1

Labos Elemer, Jul 10 2000

nonn

Also numbers k such that (10^k-7)/3 is prime.
Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.
The history of the discovery of these numbers may be as follows: a(1)-a(7), Makowski; a(8)-a(18), Caldwell; a(19), Earls; a(20)-a(31), Kamada. (Corrections to this account will be welcomed.)
Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos
No more than 14 consecutive exponents can provide primes because for exponents 15m+2, 16m+9, 18m+12, 22m+21, terms are divisible by 31, 17, 19, 23 respectively. Here 7 of possible 14 is realized. - Labos Elemer, Jan 19 2005
(10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - Seiichi Manyama, Nov 05 2016

C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.

A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp. 61:29-68, 1993.
Makoto Kamada, Prime numbers of the form 33...331.
Mathematics.StackExchange.com, 31,331,3331, 33331,333331,3333331,33333331 are prime
F. Morain, Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm, INRIA Research Report, # 911, October 1988.
Dave Rusin, Primes in exponential sequences [Broken link]
Dave Rusin, Primes in exponential sequences [Cached copy]
Index entries for primes involving repunits

Cf. A051200, A033175, A055520.

Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
One may run the prime certificate program as follows <True]}, {n, 1, 16}] (* Labos Elemer *)

for(n=1,2000, if(isprime((10^n-7)/3),print(n)))

a(n) = A055520(n) + 1.

Corrected and extended by Jason Earls, Sep 22 2001
a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.
a(32)-a(33) from Leonid Durman, Jan 09-10 2012
a(34)-a(35) from Kamada data by Tyler Busby, Apr 14 2024

A104484 Number of distinct prime divisors of 33...331 (with n 3s).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 2, 2, 3, 1, 3, 4, 3, 2, 4, 3, 3, 2, 4, 3, 3, 2, 3, 4, 6, 2, 2, 3, 3, 4, 4, 1, 3, 3, 6, 6, 3, 2, 3, 5, 3, 1, 3, 3, 4, 6, 3, 3, 5, 3, 5, 1, 3, 6, 5, 4, 3, 5, 3, 3, 5, 4, 6, 6, 3, 6, 2, 2, 4, 1, 5, 5, 5, 4, 4, 7, 2, 2, 5, 6

Offset: 0

Parthasarathy Nambi, Apr 18 2005

Interestingly, the first seven members in this sequence are all primes.

			Number of distinct prime divisors of 31 is 1 (prime).
Number of distinct prime divisors of 331 is 1 (prime).
Number of distinct prime divisors of 3331 is 1 (prime).
Number of distinct prime divisors of 33331 is 1 (prime).
Number of distinct prime divisors of 333331 is 1 (prime).
Number of distinct prime divisors of 3333331 is 1 (prime).
Number of distinct prime divisors of 33333331 is 1 (prime).

Amiram Eldar, Table of n, a(n) for n = 0..203

Cf. A001221, A033175, A051200, A105267.

Table[Length[FactorInteger[(10^(n + 1) - 7)/3]], {n, 1, 50}] (* Stefan Steinerberger *)

a(n) = omega((10^(n + 1) - 7)/3); \\ Michel Marcus, May 13 2020

a(n) = A001221(A033175(n+1)). - Amiram Eldar, Jan 24 2020, corrected, May 13 2020

More terms from Amiram Eldar, Jan 24 2020

a(0) inserted by Amiram Eldar, May 13 2020

A055520 Numbers k such that 30*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 17, 39, 49, 59, 77, 100, 150, 318, 381, 783, 1731, 1917, 8854, 11244, 11959, 12129, 18532, 22717, 23364, 24252, 24548, 25323, 30177, 53717, 380975, 424860

Offset: 1

Eric W. Weisstein

Also numbers k such that (10^(k+1)-7)/3 is prime.

Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, p. 194 (1997).

Makoto Kamada, Prime numbers of the form 33...331.
Eric Weisstein's World of Mathematics, 3
Index entries for primes involving repunits

Cf. A055557.

Indices of A033175 that are primes. Cf. A051200, A055557.

Do[ If[ PrimeQ[30*(10^n - 1)/9 + 1], Print[n]], {n, 0, 50410}]

a(n) = A055557(n) - 1. - Robert Price, Jan 30 2015

a(32)-a(33) from Leonid Durman, Jan 09-10 2012

A091189 Primes of the form 20*R_k + 1, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

2221, 222222222222222221, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221

Offset: 1

Rick L. Shepherd, Feb 22 2004

Primes of the form 222...221.

The number of 2's in each term is given by the corresponding term of A056660 and so the first term too large to include above is 222...2221 (with 120 2's).

Makoto Kamada, Prime numbers of the form 22...221.
Index entries for primes involving repunits

Cf. A056660 (corresponding k), A084832.

Cf. A051200, A004022, A093176, A093177, A089346.

A092675 Primes of the form 80*R_k + 1, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

881, 8888888888888888881, 8888888888888888888888888888888888888888888888888888888888888888888888888888881

Offset: 1

Rick L. Shepherd, Mar 02 2004

Primes of the form 888...881.
The number of 8's in each term is given by the corresponding term of A056664 and so the first term too large to include above is 888...8881 (with 138 8's).
Primes of the form (8*10^k - 71)/9. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 88...881.
Index entries for primes involving repunits

Cf. A056664 (corresponding k).

Cf. A051200, A004022, A093176, A093177, A089346, A093174, A092571.

Select[Table[10 FromDigits[PadRight[{},n,8]]+1,{n,150}],PrimeQ] (* Harvey P. Dale, Aug 07 2019 *)

A105267 a(n) = the number of divisors of 33...31, with n 3s.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 8, 8, 16, 4, 4, 8, 2, 8, 24, 8, 4, 16, 8, 8, 4, 16, 8, 8, 6, 8, 16, 64, 4, 4, 8, 8, 16, 16, 2, 8, 8, 64, 64, 8, 4, 8, 32, 8, 2, 8, 8, 16, 64, 8, 8, 32, 8, 32, 2, 8, 64, 32, 16, 8, 32, 8, 8, 32, 16, 64, 64, 8, 64, 4, 4, 16, 2

Offset: 0

Parthasarathy Nambi, Apr 29 2005

nonn

The first seven 33...31 numbers are prime, so those terms are 2. - Don Reble, Oct 26 2006

Amiram Eldar, Table of n, a(n) for n = 0..203

Cf. A000005, A051200, A033175, A104484.

a[n_] := DivisorSigma[0, (10^(n + 1) - 7)/3]; Array[a, 30, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = numdiv((10^(n + 1) - 7)/3); \\ Michel Marcus, May 13 2020

a(n) = A000005(A033175(n)). - Amiram Eldar, May 13 2020

More terms from Don Reble, Oct 26 2006

A105427 Numbers n such that the near-repdigit number consisting of a 1 followed by n 3's (i.e., of form 1333...33) is composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Lekraj Beedassy, Apr 08 2005

nonn

Complement of A056698.

FactorDB, Factorizations of (10^n-1)/3+10^n
Makoto Kamada, Factorizations of 133...33.

Cf. A051200, A089017, A093671.

Select[Range[100],CompositeQ[FromDigits[PadRight[{1},#,3]]]&]-1 (* Harvey P. Dale, Jul 23 2014 *)

isok(n) = ! isprime(10^n+(10^n-1)/3) \\ Michel Marcus, Jul 28 2013

A123568 Prime numbers of the form (10^n - 7)/3.

Original entry on oeis.org

31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

Artur Jasinski, Nov 12 2006

nonn

The number of initial 3s is n - 1.

Note that each n from 2 to 8 gives primes, but after that the n that correspond to primes are progressively further apart. Singh (1997) gives this as an example of why mathematicians don't trust a preponderance of evidence as proof: in the 17th century, when factoring numbers with as few as eight digits wasn't as easy as it is today, the pattern suggested that all numbers of this form are prime. - Alonso del Arte, Nov 11 2012

			a(7) = 33333331 because that is the seventh number of the specified form to be prime.
333333331 is not in the sequence because it is composite, being the product of 17 and 19607843.

Simon Singh, Fermat's Enigma. New York: Walker & Company (1997) p. 159.

Alois P. Heinz, Table of n, a(n) for n = 1..17

Cf. A055557, A051200, A033175, A065586, A068104.

Do[If[PrimeQ[(10^n - 7)/3], Print[(10^n - 7)/3]], {n, 1, 100}] (* Jasinski *)
Select[(10^Range[50] - 7)/3, PrimeQ[#] &] (* Alonso del Arte, Nov 11 2012 *)
Select[Table[FromDigits[PadLeft[{1},n,3]],{n,50}],PrimeQ] (* Harvey P. Dale, Dec 05 2018 *)

select(ispseudoprime, vector(20, n, (10^n-7)/3)) \\ Charles R Greathouse IV, Nov 12 2012

cp's OEIS Frontend

A033175 n 3's followed by 1.

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A055558 Primes of the form 1999...999.

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A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A104484 Number of distinct prime divisors of 33...331 (with n 3s).

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A055520 Numbers k such that 30*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A091189 Primes of the form 20*R_k + 1, where R_k is the repunit (A002275) of length k.

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A092675 Primes of the form 80*R_k + 1, where R_k is the repunit (A002275) of length k.

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A105267 a(n) = the number of divisors of 33...31, with n 3s.

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