A004022 -id:A004022 - 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

A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

Original entry on oeis.org

7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003

Primes that are base-b repunits with three or more digits for at least one b >= 2: Primes in A053696. Subsequence of A000668 U A076481 U A086122 U A165210 U A102170 U A004022 U ... (for each possible b). - Rick L. Shepherd, Sep 07 2009
From Bernard Schott, Dec 18 2012: (Start)
Also known as Brazilian primes. The primes that are not Brazilian primes are in A220627.
The number of terms k+1 is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3*37.
The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33 (see Theorem 4 of Quadrature article in the Links). (End)
It is not known whether there are infinitely many Brazilian primes. See A002383. - Bernard Schott, Jan 11 2013
Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - Thomas Ordowski, Apr 25 2013
Number of terms less than 10^n: 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ..., . - Robert G. Wilson v, Mar 31 2014
From Bernard Schott, Apr 08 2017: (Start)
Brazilian primes fall into two classes:
1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,
2) when n is composite, we get sequence A285017. (End)
The conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link Bernard Schott, Quadrature, Conjecture 1, page 36) is false. Thanks to Giovanni Resta, who found that a(856) = 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73 is the 141385th Sophie Germain prime. - _Bernard Schott, Mar 08 2019

			13 is a term since it is prime and 13 = 1 + 3 + 3^2 = 111_3.
31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = 11111_2.
From _Hartmut F. W. Hoft_, May 08 2017: (Start)
The sequence represented as a sparse matrix with the k-th column indexed by A006093(k+1), primes minus 1, and row n by A000027(n+1). Traversing the matrix by counterdiagonals produces a non-monotone ordering.
    2    4      6        10             12          16
2  7    31     127      -              8191        131071
3  13   -      1093     -              797161      -
4  -    -      -        -              -           -
5  31   -      19531    12207031       305175781   -
6  43   -      55987    -              -           -
7  -    2801   -        -              16148168401 -
8  73   -      -        -              -           -
9  -    -      -        -              -           -
10  -    -      -        -              -           -
11  -    -      -        -              -           50544702849929377
12  157  22621  -        -              -           -
13  -    30941  5229043  -              -           -
14  211  -      8108731  -              -           -
15  241  -      -        -              -           -
16 -    -      -        -              -           -
17  307  88741  25646167 2141993519227  -           -
18  -    -      -        -              -           -
19  -    -      -        -              -           -
20  421  -      -        10778947368421 -           689852631578947368421
21  463  -      -        17513875027111 -           1502097124754084594737
22  -    245411 -        -              -           -
23  -    292561 -        -              -           -
24  601  346201 -        -              -           -
Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:
column  2:  A002383            row 2:  A000668
column  4:  A088548            row 3:  A076481
column  6:  A088550            row 4:  -
column 10:  A162861            row 5:  A086122.
(End)

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10831 (terms up to 10^10; terms 1..3880 from T. D. Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A189891 (complement), A125134 (Brazilian numbers), A220627 (Primes that are non-Brazilian).
Cf. A003424 (n restricted to prime powers).
Cf. A053696, A086930, A059055.
Equals A023195 \3 Union A285017 with empty intersection.
Primes of the form (b^k-1)/(b-1) for b=2: A000668, b=3: A076481, b=5: A086122, b=6: A165210, b=7: A102170, b=10: A004022.
Primes of the form (b^k-1)/(b-1) for k=3: A002383, k=5: A088548, k=7: A088550, k=11: A162861.

a085104 n = a085104_list !! (n-1)
a085104_list = filter ((> 1) . a088323) a000040_list
-- Reinhard Zumkeller, Jan 22 2014

max = 140; maxdata = (1 - max^3)/(1 - max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 - m^i)/(1 - m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)
f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[ id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Select[ Range[2, 60], 1 + f@# != Prime@# &] (* Robert G. Wilson v, Mar 31 2014 *)

list(lim)=my(v=List(),t,k);for(n=2,sqrt(lim), t=1+n;k=1; while((t+=n^k++)<=lim,if(isprime(t), listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 08 2013

A085104_vec(N,L=List())=forprime(K=3,logint(N+1,2),for(n=2,sqrtnint(N-1,K-1),isprime((n^K-1)\(n-1))&&listput(L,(n^K-1)\(n-1))));Set(L) \\ M. F. Hasler, Jun 26 2018

A010051(a(n)) * A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

More terms from David Wasserman, Jan 26 2005

A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111

Offset: 1

N. J. A. Sloane

From Bill Gosper, Jan 24 2003, in a posting to the Math Fun Mailing List: (Start)
Recall Sloane's old request for more terms of A003459 = (2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 199 311 337 373 733 919 991 ...) and Richard C. Schroeppel's astonishing observation that the next term is 1111111111111111111. Absent Rich's analysis, trying to extend this sequence makes a great set of beginner's programming exercises. We may restrict the search to combinations of the four digits 1,3,7,9, only look at starting numbers with nondecreasing digits, generate only unique digit combinations, and only as needed. (We get the target sequence afterward by generating and merging the various permutations, and fudging the initial 2,3,5,7.)
To my amazement the (uncompiled, Macsyma) program printed 11,13,...,199,337, and after about a minute, 1111111111111111111!
And after a few more minutes, (10^23-1)/9! (End)
Boal and Bevis say that Johnson (1977) proves that if there is a term > 1000 with exactly two distinct digits then it must have more than nine billion digits. - N. J. A. Sloane, Jun 06 2015
Some authors require permutable or absolute primes to have at least two different digits. This produces the subsequence A129338. - M. F. Hasler, Mar 26 2008
See A039986 for a related problem with more sophisticated (PARI) code (iteration over only inequivalent digit permutations). - M. F. Hasler, Jul 10 2018

Richard C. Schroeppel, personal communication.
Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 20-21.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977. [Related paper, but primarily concerned with A023107 and A103443. - _N. J. A. Sloane_, Jun 06 2015]
T. N. Bhargava and P. H. Doyle, On the existence of absolute primes, Math. Mag., 47 (1974), 233.
J. L. Boal and J. H. Bevis, Permutable primes. Math. Mag., 55 (No. 1, 1982), 38-41.
C. Caldwell, The prime glossary: Permutable Prime.
J. P. Delahaye, Persistent Primes, Illustrating Permutable, Circular, Right & Left Truncatable Primes, Pour La Science no 256.
James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.
A. W. Johnson, Absolute primes, Mathematics Magazine, 1977, vol. 50, pp. 100-103.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
W. Schneider, MATHEWS, Circular, Permutable, Truncatable and Deletable Primes.
A. Slinko, Absolute Primes Oct. 2000.
A. Slinko, Absolute Primes, Oct. 2000 [Cached copy, permission requested].
Wikipedia, Permutable prime.
Index entries for sequences related to truncatable primes.

Includes all of A004022 = A002275(A004023).
A258706 gives minimal representatives of the permutation classes.
Cf. A010051, A023107, A016114, A103443, A129338, A141263.
Cf. A039986.

import Data.List (permutations)
a003459 n = a003459_list !! (n-1)
a003459_list = filter isAbsPrime a000040_list where
   isAbsPrime = all (== 1) . map (a010051 . read) . permutations . show
-- Reinhard Zumkeller, Sep 15 2011

f[n_]:=Module[{b=Permutations[IntegerDigits[n]],q=1},Do[If[!PrimeQ[c=FromDigits[b[[m]]]],q=0;Break[]],{m,Length[b]}];q];Select[Range[1000],f[#]>0&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2011 *)
(* Linear complexity: can't reach R(19). See A258706. - Bill Gosper, Jan 06 2017 *)

for(n=1, oo, my(S=[],r=10^n\9); for(a=1, 9^(n>1), for(b=if(n>2, 1-a), 9-a, for(j=0, if(b, n-1), ispseudoprime(a*r+b*10^j)||next(2)); S=concat(S,vector(if(b,n,1),k,a*r+10^(k-1)*b))));apply(t->printf(t","),Set(S))) \\ M. F. Hasler, Jun 26 2018

Conjecture: for n >= 23, a(n) = A004022(n-21). - Max Alekseyev, Oct 08 2018

The next terms are a(25)=A002275(317), a(26)=A002275(1031), a(27)=A002275(49081).

A032758 Undulating primes (digits alternate).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1212121, 1616161, 323232323, 383838383

Offset: 1

Patrick De Geest, May 15 1998

Sometimes called "smoothly undulating primes", to distinguish them from A059168.

C. A. Pickover, "Keys to Infinity", Wiley 1995, p. 159,160.
C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

Michael S. Branicky, Table of n, a(n) for n = 1..131 (terms 1..100 from Sean A. Irvine)
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Charles W. Trigg, Nine-digit patterned palindromic primes, Crux Mathematicorum, Vol. 7, No. 6, June - July 1981, pp. 168-170.

Cf. A033619, A030291, A059168, A242541, A004022.

a[n_] := DeleteDuplicates[Take[IntegerDigits[n],{1,-1,2}]]; b[n_] := DeleteDuplicates[Take[IntegerDigits[n],{2,-1,2}]]; t={}; Do[p=Prime[n]; If[p<10, AppendTo[t,p], If[Length[a[p]] == Length[b[p]] == 1 && a[p][[1]] != b[p][[1]], AppendTo[t,p]]], {n,3*10^7}]; t (* Jayanta Basu, May 04 2013 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    yield from (p for p in primerange(2, 100) if p != 11)
    yield from (t for t in (int((A+B)*d2+A) for d2 in count(1) for A in "1379" for B in "0123456789" if A != B) if isprime(t))
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 09 2022

Sequence corrected by Juri-Stepan Gerasimov, Jan 28 2010

Offset corrected by Arkadiusz Wesolowski, Sep 13 2011

A031974 1 repeated prime(n) times.

Original entry on oeis.org

11, 111, 11111, 1111111, 11111111111, 1111111111111, 11111111111111111, 1111111111111111111, 11111111111111111111111, 11111111111111111111111111111, 1111111111111111111111111111111, 1111111111111111111111111111111111111

Offset: 1

J. Castillo (arp(AT)cia-g.com) [Broken email address?]

Salomaa's first example of an infinite language. - N. J. A. Sloane, Dec 05 2012
If p is a prime and gcd(p,b-1)=1, then (b^p-1)/(b-1) == 1 (mod p); by Fermat's little theorem. For example 1111111 == 1 (mod 7). - Thomas Ordowski, Apr 09 2016
Also Mersenne numbers (A001348) written in binary. - Kritsada Moomuang, May 13 2025

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 2. - From N. J. A. Sloane, Dec 05 2012

N. J. A. Sloane, Table of n, a(n) for n = 1..50
Fanel Iacobescu, Smarandache Partition Type Sequences, in Bulletin of Pure and Applied Sciences, India, Vol. 16E, No. 2, 1997, pp. 237-240
M. Le and K. Wu, The Primes in the Smarandache Unary Sequence, Smarandache Notions Journal, Vol. 9, No. 1-2. 1998, 98-99.
Eric Weisstein's World of Mathematics, Smarandache Sequences

A004022 is the subsequence of primes. - Jeppe Stig Nielsen, Sep 14 2014

Cf. A001348.

[(10^p-1)/9: p in PrimesUpTo(40)]; // Vincenzo Librandi, May 29 2014

f:=n->(10^ithprime(n)-1)/9; [seq(f(n),n=1..20)]; # N. J. A. Sloane, Dec 05 2012

Table[FromDigits[PadRight[{},Prime[n],1]],{n,15}] (* Harvey P. Dale, Apr 10 2012 *)

a(n) = A000042(A000040(n)). - Jason Kimberley, Dec 19 2012

a(n) = (10^prime(n) - 1)/9. - Vincenzo Librandi, May 29 2014

More terms from Erich Friedman

Corrected and extended by Harvey P. Dale, Apr 10 2012

A065584 Smallest prime beginning with exactly n 1's.

Original entry on oeis.org

2, 13, 11, 1117, 11113, 111119, 11111101, 11111117, 111111113, 11111111129, 11111111113, 1111111111139, 11111111111123, 1111111111111013, 1111111111111123, 11111111111111101, 11111111111111119

Offset: 0

Robert G. Wilson v, Nov 28 2001

Largest terms are only probable primes. Some terms of the sequence are also in A004022 (repunit primes) or A056710 (all digits same except last digit). All terms of A004022 are in the current sequence.

According to the Magma Calculator (at http://magma.maths.usyd.edu.au/calc/), all 201 terms in the table (and thus all 17 terms listed above) are, in fact, prime. - Jon E. Schoenfield, Aug 24 2009

M. F. Hasler, Table of n, a(n) for n = 0..200

Cf. A065821, A068103, A068104, A068105.

Cf. A004022 (repunit primes), A056710 (near-repdigit primes).

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A107689 Primes with digital product = 3.

Original entry on oeis.org

3, 13, 31, 113, 131, 311, 11113, 11131, 11311, 113111, 131111, 311111, 11111131, 11111311, 11113111, 11131111, 111111113, 111111131, 111113111, 131111111, 11111111113, 11111111131, 11113111111, 11131111111, 31111111111

Offset: 1

Zak Seidov and Robert G. Wilson v, May 20 2005

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A004022, A107612, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

Subsequence of A034050.

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{3, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 12}]]

from sympy import isprime
def agen():
  digits = 0
  while True:
    for i in range(digits+1):
      t = int("1"*(digits-i) + "3" + "1"*i)
      if isprime(t): yield t
    digits += 1
g = agen()
print([next(g) for i in range(25)]) # Michael S. Branicky, Mar 13 2021

A336826 Bogotá numbers: numbers k such that k = m*p(m) where p(m) is the digital product of m.

Original entry on oeis.org

0, 1, 4, 9, 11, 16, 24, 25, 36, 39, 42, 49, 56, 64, 75, 81, 88, 93, 96, 111, 119, 138, 144, 164, 171, 192, 224, 242, 250, 255, 297, 312, 336, 339, 366, 378, 393, 408, 422, 448, 456, 488, 497, 516, 520, 522, 525, 564, 575, 648, 696, 704, 738, 744, 755, 777, 792

Offset: 1

Sean A. Irvine, Aug 05 2020

Named Bogotá numbers by Tomás Uribe and Juan Pablo Fernández based on similarity of the construction to the Colombian numbers (A003052).
Some questions about these numbers:
(i) Some Bogotá numbers occur in pairs (such as 24 and 25). Are there infinitely many such pairs?
(ii) More generally, can arbitrarily long sets of consecutive numbers be found all of which are Bogotá numbers?
(iii) Can the gap between two consecutive Bogotá numbers be arbitrarily large? Answer: Yes.
From David A. Corneth, Aug 06 2020: (Start)
The only primes in this sequence are A004022.
To see if a number is a Bogotá number, we only have to look at its 7-smooth divisors. Proof: If a number k is a Bogotá number then k = m*p(m) where p(m) is 7-smooth as it's a product of digits. Furthermore, if k = m*p(m) then p(m) | k. Q.e.d. Below is an example using this idea.
To find Bogotá numbers k up to N we can make a list of 7-smooth numbers up to sqrt(N) and list the factorizations into single-digit numbers of each of these 7-smooth numbers that when concatenated give m such that m * p(m) = k where p(m) is that 7-smooth number.
For example, 10 is a 7-smooth number. Its factorizations into single-digit numbers are 2*5, 5*2, 1*2*5 and so on. This tells us that 10*25 = 250, 10*52 = 520, 10*125 = 1250 all are Bogotá numbers.
Similarily we can find odd Bogotá numbers to then find consecutive Bogotá numbers (See A336864). (End)

			From _David A. Corneth_, Aug 06 2020: (Start)
520 is a term because 52 * p(52) = 52 * 10 = 520.
Example using we only have to look at 7-smooth divisors:
520 is a term as its 7-smooth divisors d are 1, 2, 4, 5, 8, 10, 20, 40. values 520/d are 520, 260, 130, 104, 65, 52, 26, 13 of which 52 * (5*2) = 520 where (5*2) are the products of 52. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)
Math Stackexchange, Gaps between Bogotá numbers, 2020.
Puzzling Stackexchange, Pairs of Bogotá numbers, 2020.

Cf. A003052, A004022, A007954, A098736, A336864, A336876.

f(n) = vecprod(digits(n))*n; \\ A098736
isok(n) = my(k=0); for (k=0, n, if (f(k) == n, return(1))); \\ Michel Marcus, Aug 06 2020

is(n) = { my(f = factor(n), s7 = 1, d, sl = sqrtint(n)); for(i = 1, #f~, if(f[i, 1] > 7, break ); s7 *= f[i, 1]^f[i, 2]; ); d = divisors(s7); for(i = 1, #d, if(d[i] > sl, return(0)); if(n/d[i] * vecprod(digits(n/d[i])) == n, return(1); ) ); 0 } \\ David A. Corneth, Aug 06 2020

A105992 Near-repunit primes.

Original entry on oeis.org

101, 113, 131, 151, 181, 191, 211, 311, 811, 911, 1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111, 10111, 11113, 11117, 11119, 11131, 11161, 11171, 11311, 11411, 16111, 101111, 111119, 111121, 111191, 111211, 111611, 112111, 113111, 131111, 311111, 511111

Offset: 1

Shyam Sunder Gupta, Apr 29 2005

According to the prime glossary "a near-repunit prime is a prime all but one of whose digits are 1." This would also include {2, 3, 5, 7, 13, 17, 19, 31, 41, 61 and 71}, but this sequence only lists terms with more than two digits. - M. F. Hasler, Feb 10 2020

			a(2)=113 is a term because 113 is a prime and all digits are 1 except one.

C. Caldwell and H. Dubner, "The near repunit primes 1(n-k-1)01(1k)," J. Recreational Math., 27 (1995) 35-41.
Heleen, J. P., "More near-repunit primes 1(n-k-1)D(1)1(k), D=2,3, ..., 9," J. Recreational Math., 29:3 (1998) 190-195.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Top 20 Near-repdigit Primes
Chris Caldwell, The Prime Glossary, Near-repunit prime

Cf. A000042, A002275, A004022, A046053, A046413, A065074, A034093, A065083, A102782, A118505.

lst = {}; Do[r = (10^n - 1)/9; Do[AppendTo[lst, DeleteCases[Select[FromDigits[Permutations[Append[IntegerDigits[r], d]]], PrimeQ], r]], {d, 0, 9}], {n, 2, 14}]; Sort[Flatten[lst]] (* Arkadiusz Wesolowski, Sep 20 2011 *)

A107612 Primes with digital product = 2.

Original entry on oeis.org

2, 211, 2111, 111121, 111211, 112111, 1111211, 1111111121, 1111211111, 1121111111, 111111211111, 111211111111, 2111111111111, 111111111111112111, 111111112111111111, 111111211111111111, 112111111111111111

Offset: 1

Zak Seidov, May 17 2005

Corresponding indices of primes in A107611. Cf. A053666, A101987.

Harvey P. Dale, Table of n, a(n) for n = 1..684

Cf. A053666, A101987, A107611.

Cf. A004022, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

for i from 0 to 30 do it:=sum(10^j, j=0..i): for k from 0 to i do if isprime(it+10^k) then printf(`%d,`, it+10^k) fi: od:od: (Sellers)

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{2, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 19}]] (* Robert G. Wilson v, May 19 2005 *)
Select[Flatten[Table[FromDigits/@Permutations[PadRight[{2},n,1]],{n,20}]],PrimeQ]//Sort (* Harvey P. Dale, May 28 2017 *)

A107612(n) = prime(A107611(n)).

More terms from Robert G. Wilson v and James Sellers, May 19 2005

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A032758 Undulating primes (digits alternate).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A031974 1 repeated prime(n) times.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A065584 Smallest prime beginning with exactly n 1's.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions