A004023 -id:A004023 - cp's OEIS frontend

A183174 Numbers k such that (10^(2k+1) - 6*10^k - 1)/3 is prime.

Original entry on oeis.org

1, 3, 7, 61, 90, 92, 269, 298, 321, 371, 776, 1567, 2384, 2566, 3088, 5866, 8051, 9498, 12635, 24512, 32521, 43982

Offset: 1

Ray Chandler, Dec 28 2010

a(23) > 10^5. - Robert Price, Jan 29 2016

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's).
Makoto Kamada, Prime numbers of the form 33...33133...33.
Index entries for primes involving repunits.

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Do[If[PrimeQ[(10^(2n + 1) - 6*10^n - 1)/3], Print[n]], {n, 3000}]

for(n=1,1e3,if(ispseudoprime((10^(2*n+1)-6*10^n-1)/3),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077775(n) - 1)/2.

a(21)-a(22) from Robert Price, Jan 29 2016

A204940 Numbers n such that (23^n - 1)/22 is prime.

Original entry on oeis.org

5, 3181, 61441, 91943, 121949, 221411

Offset: 1

Robert Price, Jan 20 2012

No other terms < 100000.

H. Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A128000, A098438, A128002, A128003, A128004, A128005.

Select[Prime[Range[100]], PrimeQ[(23^#-1)/22]&]

is(n)=ispseudoprime((23^n-1)/22) \\ Charles R Greathouse IV, Jun 13 2017

a(5)=121949 corresponds to a probable prime discovered by Paul Bourdelais, Oct 19 2017

a(6)=221411 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A036229 Smallest n-digit prime containing only digits 1 or 2 or -1 if no such prime exists.

Original entry on oeis.org

2, 11, 211, 2111, 12211, 111121, 1111211, 11221211, 111112121, 1111111121, 11111121121, 111111211111, 1111111121221, 11111111112221, 111111112111121, 1111111112122111, 11111111111112121, 111111111111112111, 1111111111111111111, 11111111111111212121

Offset: 1

G. L. Honaker, Jr.

It is conjectured that such a prime always exists.

a(2), a(19), a(23), etc. are the prime repunits (A004023). a(1000) = (10^n-1)/9 + 111011000010.

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (terms n=1..400 from Alois P. Heinz)
Robert G. Wilson v, Comments and first 100 terms

Cf. A036937, A068086.

Do[p = (10^n - 1)/9; k = 0; While[ ! PrimeQ[p], k++; p = FromDigits[ PadLeft[ IntegerDigits[ k, 2], n] + 1]]; Print[p], {n, 1, 20}]
Table[Min[Select[ FromDigits/@Tuples[{1,2},n],PrimeQ]],{n,20}] (* Harvey P. Dale, Feb 05 2014 *)

from sympy import isprime
def A036229(n):
    k, r, m = (10**n-1)//9, 2**n-1, 0
    while m <= r:
        t = k+int(bin(m)[2:])
        if isprime(t):
            return t
        m += 1
    return -1 # Chai Wah Wu, Aug 18 2021

Edited by N. J. A. Sloane and Robert G. Wilson v, May 03 2002

Escape clause added by Chai Wah Wu, Aug 18 2021

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

A034388 Smallest prime containing at least n consecutive identical digits.

Original entry on oeis.org

2, 11, 1117, 11113, 111119, 2999999, 11111117, 111111113, 1777777777, 11111111113, 311111111111, 2111111111111, 17777777777777, 222222222222227, 1333333333333333, 11111111111111119, 222222222222222221, 1111111111111111111, 1111111111111111111

Offset: 1

Erich Friedman and N. J. A. Sloane

For n in A004023, a(n) = A002275(n). For all other n > 1, a(n) has at least n+1 digits and is (for small n) often of the form a*10^n + b*(10^n-1)/9 or a*(10^n-1)/9*10 + b, with 1 <= a <= 9 and b in {1, 3, 7, 9}. However, for n = 24, 46, 48, 58, 60, 64, 66, ..., more digits are required. Only then a(n) can have a digit 0, and if it has, '0' is often the repeated digit. The first indices where a(n) has more than n+2 digits are n = 208, 277, 346, ... - M. F. Hasler, Feb 25 2016; corrected by Robert Israel, Feb 26 2016

			a(1) = 2 because this is the smallest prime.
a(2) = 11 because this repunit with n=2 digits is prime.
a(3) = 1117 is the smallest prime with 3 repeated digits.
a(19) = (10^19-1)/9 = R(19) is again a repunit prime. Since all primes with 18 consecutive repeated digits have at least 19 digits, also a(18) = a(19). The same happens for a(22) = a(23).

M. F. Hasler and Robert Israel, Table of n, a(n) for n = 1..997 (1..200 from M. F. Hasler)
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.

Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037069, A037071.

Cf. A065821, A065586, A084673.

f:= proc(n) local d, k,x,y,z,xs,ys,zs,c,cands;
  for d from n do
    cands:= NULL;
    for k from 0 to d-n do
      if k = 0 then zs:= [0] else zs:= [seq(i,i=1..10^k-1,2)] fi;
      if d=n+k then xs:= [0]; ys:= [$1..9] else xs:= [$10^(d-k-n-1)..10^(d-k-n)-1]; ys:= [$0..9] fi;
      cands:= cands, seq(seq(seq(z + 10^k*y*(10^n-1)/9 + x*10^(k+n), x = xs),y=ys),z=zs);
    od;
    cands:= sort([cands]);
    for c in cands do if isprime(c) then return(c) fi od;
  od
end proc:
map(f, [$1..30]); # Robert Israel, Feb 26 2016

With[{s = Length /@ Split@ IntegerDigits@ # & /@ Prime@ Range[10^6]}, Prime@ Array[FirstPosition[s, #][[1]] &, Max@ Flatten@ s]] (* Michael De Vlieger, Aug 15 2018 *)

A034388(n)={for(d=0,9, my(L=[],k=0); for(a=0,10^d-1,a<10^k||k++; L=setunion(L,vector(10-!a,c,[a*10^n+10^n\9*(c-(a>0)),1])*10^(d-k)));for(i=1,#L,if(L[i][2]>1, L[i][1]+L[i][2]>L[i][1]=nextprime(L[i][1]),ispseudoprime(L[i][1]))&&return(L[i][1])))} \\ M. F. Hasler, Feb 28 2016

Edited by M. F. Hasler, Feb 25 2016

Edited by Robert Israel, Feb 26 2016

A003020 Largest prime factor of the "repunit" number 11...1 (cf. A002275).

Original entry on oeis.org

11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049

Offset: 2

N. J. A. Sloane

a(n) = R_n iff n is a term of A004023. - Bernard Schott, Jul 07 2022

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 40.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Factors of the Repunits 11 through R_40, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986, p. 219.

Max Alekseyev, Table of n, a(n) for n = 2..352 (terms a(2)..a(100) from T. D. Noe, derived from data from Yousuke Koide; a(101)..a(322) from Ray Chandler)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Patrick De Geest, Repunits and their Prime Factors
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Koide, Factorizations of Repunit Numbers
A. A. D. Steward, Factorization of Repunits[up to R(196)] [Broken link]
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002275, A004023, A006530, A102380.
Same as A005422 except for initial terms.
Smallest factor: A067063.

Table[Max[Transpose[FactorInteger[10^i - 1]][[1]]], {i, 2, 25}]
Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[-1,1]],{n,2,30}] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=local(p); if(n<2,n==1,p=factor((10^n-1)/9)~[1,]; p[length(p)])

a(n) = A006530(A002275(n)). - Ray Chandler, Apr 22 2017

More terms from Harvey P. Dale, Jan 17 2001

A037055 Smallest prime containing exactly n 1's.

Original entry on oeis.org

2, 13, 11, 1117, 10111, 101111, 1111151, 11110111, 101111111, 1111111121, 11111111113, 101111111111, 1111111118111, 11111111111411, 111111111116111, 1111111111111181, 11111111101111111, 101111111111111111, 1111111111111111171, 1111111111111111111, 111111111111111119111

Offset: 0

Patrick De Geest, Jan 04 1999

For n > 1, A037055 is conjectured to be identical to A084673. - Robert G. Wilson v, Jul 04 2003

a(n) = A002275(n) for n in A004023. For all other n < 900, a(n) has n+1 digits. - Robert Israel, Feb 21 2016

Robert Israel, Table of n, a(n) for n = 0..900

Cf. A084673, A065584, A065821, A037054, A034388, A036507-A036536.

Cf. A037053, A037057, A037059, A037061, A037063, A037065, A037067, A037069, A037071.

f:= proc(n) local m,d,r,x;
   r:= (10^n-1)/9;
   if isprime(r) then return r fi;
   r:= (10^(n+1)-1)/9;
   for m from n-1 to 1 by -1 do
     x:= r - 10^m;
     if isprime(x) then return x fi;
   od;
   for m from 0 to n do
     for d from 1 to 8 do
        x:= r + d*10^m;
        if isprime(x) then return x fi;
     od
   od;
   error("Needs more than n+1 digits")
end proc:
map(f, [$0..100]); # Robert Israel, Feb 21 2016

f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[b, {i, 1, n}], {x}]], c = Complement[Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[r ? Infinity, r, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 1], {n, 1, 18}]
Join[{2, 13}, Table[Sort[Flatten[Table[Select[FromDigits/@Permutations[Join[{n}, PadRight[{}, i, 1]]], PrimeQ], {n, 0, 9}]]][[1]], {i, 2, 20}]] (* Vincenzo Librandi, May 11 2017 *)

A037055(n)={my(p,t=10^(n+1)\9); forstep(k=n+1,1,-1, ispseudoprime(p=t-10^k) && return(p)); forvec(v=[[0, n], [1, 8]], ispseudoprime(p=t+10^v[1]*v[2]) && return(p))} \\ M. F. Hasler, Feb 22 2016

a(n) = the smallest prime in { R-10^n, R-10^(n-1), ..., R-10; R+a*10^b, a=1, ..., 8, b=0, 1, 2, ..., n }, where R = (10^(n+1)-1)/9 is the (n+1)-digit repunit. - M. F. Hasler, Feb 25 2016

a(n) = prime(A037054(n)). - Amiram Eldar, Jul 21 2025

More terms from Sascha Kurz, Feb 10 2003
Edited by Robert G. Wilson v, Jul 04 2003
a(0) = 2 inserted by Robert Israel, Feb 21 2016

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

Original entry on oeis.org

3, 6, 8, 12, 12, 64, 12, 48, 20, 48, 12, 256, 24, 48, 128, 192, 12, 640, 6, 384, 256, 288, 6, 2048, 96, 192, 96, 768, 96, 16384, 24, 6144, 128, 192, 384, 5120, 24, 24, 128, 6144, 48, 49152, 48, 4608, 1280, 192, 12, 16384, 48, 3072, 512, 1536, 48, 12288, 768

Offset: 1

Henry Bottomley, May 02 2002

			a(7)=12 since the divisors of 9999999 are 1, 3, 9, 239, 717, 2151, 4649, 13947, 41841, 1111111, 3333333, 9999999.

Table of n, a(n) for n = 1..352

Cf. A004023 (indices of 6's).
Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.
Cf. A046801, A005422, A102347, A046053, A057951, A007138, A003060, A059892, A085035.

DivisorSigma[0,#]&/@(10^Range[60]-1) (* Harvey P. Dale, Jan 14 2011 *)
Table[DivisorSigma[0, 10^n - 1], {n, 60}] (* T. D. Noe, Aug 18 2011 *)

a(n) = numdiv(10^n - 1); \\ Michel Marcus, Sep 08 2015

a(n) = A000005(A002283(n)).
a(n) = Sum_{d|n} A059892(d).
a(n) = A070529(n)*(A007949(n)+3)/(A007949(n)+1).

Terms to a(280) in b-file from Hans Havermann, Aug 19 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, May 04 2022

A005422 Largest prime factor of 10^n - 1.

Original entry on oeis.org

3, 11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049

Offset: 1

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 1..352
K. Beschorner, Factorizations of Repunit Numbers of the form (10^n-1)/9 (111111...). [Retrieved July 30, 2015]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Yousuke Koide, Factorizations of Repunit Numbers.
S. S. Wagstaff, Jr., The Cunningham Project

Same as A003020 except for the additional a(1) = 3.
Cf. A002275, A002283, A004023, A006530, A067063, A075024, A102380.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(10^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

A005422 := proc(n)
    10^n-1 ;
    A006530(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[FactorInteger[10^n - 1][[-1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

a(n)=vecmax(factor(10^n-1)[,1]) \\ Simplified by M. F. Hasler, Jul 30 2015

For n > 1, a(n) = A003020(n). For 1 < n < 10, a(n) = A075024(n). - M. F. Hasler, Jul 30 2015
a(n) = A006530(A002283(n)). - Vincenzo Librandi, Jul 13 2016
a(A004023(n)) = A002275(A004023(n)). - Bernard Schott, May 24 2022

Terms to a(100) in b-file from Yousuke Koide added by T. D. Noe, Dec 06 2006
Edited by M. F. Hasler, Jul 30 2015
a(101)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

A067063 Smallest prime factor of repunit(n) = (10^n-1)/9 (A002275).

Original entry on oeis.org

11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239

Offset: 2

Amarnath Murthy, Jan 03 2002

nonn

a(n) = A003020(n) = R_(n) iff n is a term of A004023. - Bernard Schott, May 22 2022

David Wells, The Penguin Dictionary of Curious and Interesting Numbers.

Ray Chandler, Table of n, a(n) for n = 2..508 (first 499 terms from T. D. Noe - corrected 7 terms)
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Koide, Factorizations of Repunit Numbers
Amarnath Murthy, On the divisors of Smarandache Unary Sequence, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000, page 184.
Samuel S. Wagstaff, the Cunningham Project

Cf. A002275, A004023, A020639, A102380.

Largest factor: A003020.

'min(op(numtheory[factorset]((10^k-1)/9)))'$k=2..50; # M. F. Hasler, Nov 21 2006

a = {}; Do[a = Append[a, FactorInteger[(10^n - 1)/9][[1, 1]]], {n, 2, 111} ]; a
Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[1,1]],{n,2,50}] (* Harvey P. Dale, Dec 10 2013 *)

a(3n) = 3, a(6n-4) = a(6n-2) = 11, a(30n-25) = a(30n-5) = 41, ... - M. F. Hasler, Nov 21 2006

a(n) = A020639(A002275(n)). - Ray Chandler, Apr 22 2017

More terms from Robert G. Wilson v, Jan 04 2002

cp's OEIS Frontend

A183174 Numbers k such that (10^(2k+1) - 6*10^k - 1)/3 is prime.

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A204940 Numbers n such that (23^n - 1)/22 is prime.

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A036229 Smallest n-digit prime containing only digits 1 or 2 or -1 if no such prime exists.

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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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A034388 Smallest prime containing at least n consecutive identical digits.

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A003020 Largest prime factor of the "repunit" number 11...1 (cf. A002275).

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A037055 Smallest prime containing exactly n 1's.

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