A001562 -id:A001562 - cp's OEIS frontend

A004023 Indices of prime repunits: numbers k such that 11...111 (with k 1's) = (10^k - 1)/9 is prime.

2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207

Offset: 1

People who search for repunit primes or repdigit primes may be looking for this entry.
The indices of primes with digital product (i.e., product of digits) equal to 1.
As of August 2014, only the first five repunits, through (10^1031-1)/9, have been proved prime. The next four repunits are known only to be probable primes and have not been proved to be prime. - Robert Baillie, Aug 17 2014
These indices p must also be prime. If p is not prime, say p = m*n, then 10^(m*n) - 1 = ((10^m)^n) - 1 => 10^m - 1 divides 10^(m*n) - 1. Since 9 divides 10^m - 1 or (10^m - 1)/9 = q, it follows q divides (10^p - 1)/9. This is a result of the identity, a^n - b^n = (a - b)(a^(n-1) + a^(n-2)*b + ... + b^(n-1)). - Cino Hilliard, Dec 23 2008
The numbers R_n = 11...111 = (10^n - 1)/9 with n in this sequence A004023, except for n = 2, are prime repunits in base ten, so they are prime Brazilian numbers belonging to A085104. [See Links: Les nombres brésiliens.] - Bernard Schott, Dec 24 2012
Search limit is 10800000, currently. - Serge Batalov, Jul 01 2021
On March 22 2022 the probable prime R49081 was proved to be a prime, and on May 15 2023 the probable prime R86453 was proved to be a prime. - Bassam Abdul-Baki, Dec 17 2024

			2 appears because the 2-digit repunit 11 is prime.
3 does not appear because 111 = 3 * 37 is not prime.
19 appears because the 19-digit repunit 1111111111111111111 is prime.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 19, pp 6, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, 1994; see p 146 problem 22.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 235.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 142857 at pp. 197-198.

Paul Bourdelais, A Generalized Repunit Conjecture, NMBRTHRY, 25 Jun 2009.
John Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
John Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Chris K. Caldwell, The Prime Pages, Top 20: Repunit (lists certified primes with n >= 1000)
Patrick De Geest, Circular Primes
Giovanni Di Maria, Repunit Primes Project
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Harvey Dubner, New probable prime repunit, R(49081), Number Theory List, Sep 09 1999.
Harvey Dubner, Repunit R49081 is a probable prime, Math. Comp., 71 (2001), 833-835.
Harvey Dubner, Posting to Number Theory List : Apr 03 2007
Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Solé, On Polynomial Pairs of Integers, arXiv:1210.7593 [math.NT], 2012-2014.
Martianus Frederic Ezerman, Bertrand Meyer and Patrick Solé, On Polynomial Pairs of Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.5.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See also Repunit Numbers, Ch. 11, 327-352.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See p. 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Henri Lifchitz, Mersenne and Fermat primes field
Tom Muller, Ist die Folge der Primzahl-quersummen beschrankt?, Elem. Math. 66 (2011) 146-154; doi:10.4171/EM/183.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Andy Steward, Prime Generalized Repunits
Sam Wagstaff, Jr., The Cunningham Project
E. Wegrzynowski, Nombres 1_[n] premiers
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Repunit Prime
H. C. Williams and Harvey Dubner, The primality of R1031, Math. Comp., 47(176), Oct 1986, 703-711.
Index entries for primes involving repunits

See A004022 for the actual primes.

Cf. A055557, A002275, A085104.

[p: p in PrimesUpTo(500) | IsPrime((10^p - 1) div 9)]; // Vincenzo Librandi, Nov 06 2014

Select[Range[271000], PrimeQ[FromDigits[PadRight[{}, #, 1]]] &] (* Harvey P. Dale, Nov 05 2011 *)
repUnsUpTo[k_] := ParallelMap[If[PrimeQ[#] && PrimeQ[(10^# - 1)/9], #, Nothing] &, Range[k]]; repUnsUpTo[5000] (* Mikk Heidemaa, Apr 24 2017 *)

forprime(x=2,20000,if(ispseudoprime((10^x-1)/9),print1(x","))) \\ Cino Hilliard, Dec 23 2008

from sympy import isprime; {print(n, end = ', ') for n in range(1, 10**7) if isprime(n) and isprime(10**n//9)} # (Note that sympy.isprime is only a pseudo-primality test.) - Ya-Ping Lu, Dec 20 2021, edited by M. F. Hasler, Mar 28 2022

a(6) = 49081 PRP found by Harvey Dubner - posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU) Sep 09, 1999; proved prime by Paul Underwood, Mar 21 2022.
a(7) = 86453 found using pfgw (a faster version of PrimeForm) on Oct 26 2000 by Lew Baxter (posting to Number Theory List), Oct 26, 2000; proved prime by Andreas Enge, May 16 2023.
a(8) = 109297 was apparently discovered independently by (in alphabetical order) Paul Bourdelais and Harvey Dubner around Mar 26-28 2007.
a(9) = 270343, was found Jul 11 2007 by Maksym Voznyy and Anton Budnyy, subsequently confirmed as a(9) (see Repunit Primes Project link) by Robert Price, Dec 14 2010
a(10) = 5794777 was found Apr 20 2021 by Ryan Propper and Serge Batalov
a(11) = 8177207 was found May 08 2021 by Ryan Propper and Serge Batalov

A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 5, 3, 3, 4, 7, 5, 4, 3, 2, 4, 8, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 9, 4, 5, 6, 4, 3, 10, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 10, 5, 6, 5, 4, 6, 3, 3, 12, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 12, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4, 5, 13

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

2^(a(2n)-1)-1 predicts the number of pair-solutions of even length L for AB = A^2 + B^2. For instance, with length 18 we have 10^18 + 1 = 101*9901*999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see reference 'Puzzle 104' for details).

Max Alekseyev, Table of n, a(n) for n = 1..331 (first 310 terms from Ray Chandler)
Makoto Kamada, Factorizations of 100...001.
Carlos Rivera, Puzzle 104, The Prime Puzzles & Problems Connection.
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): this sequence (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A003021, A001271, A046053, A001562, A057951, A119704, A344897.

PrimeOmega[10^Range[100]+1] (* Harvey P. Dale, May 02 2021 *)

a(n)=bigomega(10^n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057951(2n) - A057951(n). - T. D. Noe, Jun 19 2003

A095372 1+integers repeating "90" decimal digit pattern.

Original entry on oeis.org

1, 91, 9091, 909091, 90909091, 9090909091, 909090909091, 90909090909091, 9090909090909091, 909090909090909091, 90909090909090909091, 9090909090909090909091, 909090909090909090909091

Offset: 0

Labos Elemer, Jun 07 2004

These numbers arise for example as divisors of several repunits (A002275).
The aerated sequence A(n) = [1, 0, 91, 0, 9091, 0, 909091,...] is a divisibility sequence, i.e., A(n) divides A(m) whenever n divides m. It is the case P1 = 0, P2 = -11^2, Q = 10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Aug 22 2019
Except for a(0) = 1, these terms M are such that 21 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1. Actually 21 is A329914(1) and a(1) = A329915(1) = 91, and the terms >=91 form the set {M_21}; for example, 21 * 909091 = 1(909091)1. - Bernard Schott, Dec 01 2019

			Digit-pattern P=[ab..z] repeating integers equal formally with P*(-1+10^(Ln))/(-1+10^L), where L is the length of pattern;
a(9) divides A002275(38) repunit. See A095371.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002275, A095371.
Cf. A001562, A015585, A054416, A097209, A100047, A152577.
Cf. A329914, A329915.

Mathematica
```
Table[1+90*(100^n-1)/99, {n, 0, 20}]
```

a(n) = 1 + 90*(-1 + 100^n)/99 = (10^(2*n+1) + 1)/11. - Rick L. Shepherd, Aug 01 2004
From Colin Barker, Jul 03 2013: (Start)
a(n) = 101*a(n-1) - 100*a(n-2).
G.f.: -(10*x-1)/((x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(1 + 10*(exp(99*x) - 1)/11). - Elmo R. Oliveira, Mar 15 2025

A126856 Numbers n such that (31^n + 1)/32 is prime.

Original entry on oeis.org

109, 461, 1061, 50777

Offset: 1

Alexander Adamchuk, Mar 23 2007

All terms are primes.

a(5) > 10^5. - Robert Price, Jul 12 2013

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
Eric Weisstein's World of Mathematics, Repunit.
H. Lifchitz, Mersenne and Fermat primes field

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382. Cf. A084741, A084742, A065507, A126659.

Do[ p=Prime[n]; If[ PrimeQ[ (31^p + 1)/32 ], Print[p] ], {n,1,1100} ]

is(n)=isprime((31^n+1)/32) \\ Charles R Greathouse IV, Feb 17 2017

a(4) from Robert Price, Jul 12 2013

A229145 Numbers k such that (36^k + 1)/37 is prime.

Original entry on oeis.org

31, 191, 257, 367, 3061, 110503, 1145393

Offset: 1

Robert Price, Sep 15 2013

All such numbers k are prime.
Note that a(6) = 110503 corresponds to (36^110503 + 1)/37, which is only a probable prime with 171975 digits.
The primes corresponding to the terms of this sequence have 1 as their last digit and an even number as their next-to-last digit. - Iain Fox, Dec 08 2017

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Do[ p=Prime[n]; If[ PrimeQ[ (36^p + 1)/37 ], Print[p] ], {n, 1, 9592} ]

is(n)=isprime((36^n+1)/37) \\ Charles R Greathouse IV, Feb 17 2017

a(6) = 110503 (posted by Lelio R. Paula on primenumbers.net) from Paul Bourdelais, Dec 08 2017

a(7) from Paul Bourdelais, Nov 03 2023

A185240 Numbers k such that (35^k + 1)/36 is prime.

Original entry on oeis.org

11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623, 280979

Offset: 1

Robert Price, Aug 29 2013

All terms are primes. a(11) > 10^5.

Paul Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit

Do[ p=Prime[n]; If[ PrimeQ[ (35^p + 1)/36 ], Print[p] ], {n, 1, 9592} ]

is(n)=isprime((35^n+1)/36) \\ Charles R Greathouse IV, Feb 17 2017

a(11)=135623 found as probable prime and added by Paul Bourdelais, Jul 05 2018

a(12) from Paul Bourdelais, Sep 13 2021

A229524 Numbers k such that (38^k + 1)/39 is prime.

Original entry on oeis.org

5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591

Offset: 1

Robert Price, Sep 25 2013

All terms are primes. a(9) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Do[ p=Prime[n]; If[ PrimeQ[ (38^p + 1)/39 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((38^n+1)/39) \\ Charles R Greathouse IV, Feb 17 2017

a(9)=131591 corresponds to a probable prime discovered by Paul Bourdelais, Jul 03 2018

A229663 Numbers n such that (40^n + 1)/41 is prime.

Original entry on oeis.org

53, 67, 1217, 5867, 6143, 11681, 29959

Offset: 1

Robert Price, Sep 27 2013

All terms are primes.

a(8) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Do[ p=Prime[n]; If[ PrimeQ[ (40^p + 1)/41 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((40^n+1)/41) \\ Charles R Greathouse IV, Feb 17 2017

A230036 Numbers n such that (39^n + 1)/40 is prime.

Original entry on oeis.org

3, 13, 149, 15377

Offset: 1

Robert Price, Oct 05 2013

All terms are primes.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 (numbers n such that (2^n + 1)/3 is prime).

Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524.

Do[ p=Prime[n]; If[ PrimeQ[ (39^p + 1)/40 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((39^n+1)/40) \\ Charles R Greathouse IV, Feb 17 2017

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A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

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A004023 Indices of prime repunits: numbers k such that 11...111 (with k 1's) = (10^k - 1)/9 is prime.

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A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

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A095372 1+integers repeating "90" decimal digit pattern.

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A126856 Numbers n such that (31^n + 1)/32 is prime.

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A229145 Numbers k such that (36^k + 1)/37 is prime.

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A185240 Numbers k such that (35^k + 1)/36 is prime.

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A229524 Numbers k such that (38^k + 1)/39 is prime.

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