A002275 -id:A002275

A056654

Numbers k such that 10*R_k + 3 is prime, where R_k is the repunit (A002275) of length k.

+20
507

0, 1, 2, 4, 8, 10, 23, 83, 220, 1313, 2951, 20015, 51053

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Also numbers k such that (10^(k+1)+17)/9 is prime.

a(14) > 10^5. - Robert Price, Nov 01 2014

LINKS

Table of n, a(n) for n=1..13.

Makoto Kamada, Prime numbers of the form 11...113.

Index entries for primes involving repunits

FORMULA

a(n) = A097683(n+1) - 1. - Robert Price, Nov 01 2014

EXAMPLE

8 is a term because 111111113 is a prime.

MATHEMATICA

Do[ m = n; If[ primeQ[ 10*(10^n - 1)/9 + 3 ], Print[ n ] ], {n, 1, 1350} ]

PROG

(PARI) is(n)=ispseudoprime(10^n\9*10+3) \\ Charles R Greathouse IV, Nov 10 2021

CROSSREFS

Cf. A093011 (corresponding primes), A097683.

KEYWORD

hard,nonn

AUTHOR

Robert G. Wilson v, Aug 09 2000

EXTENSIONS

a(11) (only a probable prime) from Rick L. Shepherd, Mar 14 2004

a(12)-a(13) derived from A097683 by Robert Price, Nov 01 2014

STATUS

approved

A242614

Triangle read by rows: row n contains numbers with sum of digits = n, and not greater than the n-th repunit (cf. A007953 and A002275).

+20
26

0, 1, 2, 11, 3, 12, 21, 30, 102, 111, 4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, 1012, 1021, 1030, 1102, 1111, 5, 14, 23, 32, 41, 50, 104, 113, 122, 131, 140, 203, 212, 221, 230, 302, 311, 320, 401, 410, 500, 1004, 1013, 1022

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Number of terms in row n = A242622(n);

T(n,1) = A051885(n);

T(n,A242622(n)) = A002275(n);

for n > 0: number of repdigit terms in row n = A242627(n).

LINKS

Reinhard Zumkeller, Rows n = 0..8 of table, flattened

EXAMPLE

The triangle begins:

. 0: 0

. 1: 1

. 2: 2,11

. 3: 3,12,21,30,102,111

. 4: 4,13,22,31,40,103,112,121,130,202, . . . ,1021,1030,1102,1111

. 5: 5,14,23,32,41,50,104,113,122,131, . . . ,11021,11030,11102,11111 .

MATHEMATICA

Join[{0}, Flatten[Table[Select[Range[FromDigits[PadRight[{}, n, 1]]], Total[ IntegerDigits[ #]] == n&], {n, 5}]]] (* Harvey P. Dale, Oct 08 2019 *)

PROG

(Haskell)

a242614 n k = a242614_row n !! (k-1)

a242614_row n = filter ((== n) . a007953) [n .. a002275 n]

a242614_tabf = map a242614_row [0..]

CROSSREFS

Cf. A007953 (sum of digits), A002275 (repunits).

Cf. A011557, A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223, A052224, A166311, A235151, A143164, A235225, A235226, A235227, A166370, A235228, A166459, A235229.

KEYWORD

nonn,tabf,base

AUTHOR

Reinhard Zumkeller, Jul 16 2014

STATUS

approved

A003020

Largest prime factor of the "repunit" number 11...1 (cf. A002275).
(Formerly M4790)

+20
21

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

REFERENCES

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.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A002275, A004023, A006530, A102380.

Same as A005422 except for initial terms.

Smallest factor: A067063.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Harvey P. Dale, Jan 17 2001

STATUS

approved

A096508

Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

+20
20

2, 14, 17, 35, 4175, 4472, 9812, 12260, 12341, 13760, 14576, 53411, 144683, 148328

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also numbers k such that (8*10^k + 1)/9 is prime.

a(15) > 2*10^5. - Robert Price, Sep 06 2014

LINKS

Table of n, a(n) for n=1..14.

Makoto Kamada, Prime numbers of the form 88...889.

Index entries for primes involving repunits

FORMULA

a(n) = A056663(n) + 1.

EXAMPLE

35 is a term because 88888888888888888888888888888888889 (34 8's) is a prime number.

MAPLE

select(n -> isprime((8*10^n+1)/9), [$1..10000]); # Robert Israel, Sep 07 2014

MATHEMATICA

Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 0, 30000}] (* Robert G. Wilson v, Oct 15 2004 *)

PROG

(PARI)

for(n=1, 10^4, if(ispseudoprime(8*(10^n-1)/9+1), print1(n, ", "))) \\ Derek Orr, Sep 06 2014

CROSSREFS

Cf. A002275, A056663, A096503-A096507.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 12 2004

EXTENSIONS

Four missing terms (9812, 12260, 12341, 13760) added, and a(12)-a(14) added from Kamada data, by Robert Price, Sep 06 2014

STATUS

approved

A095714

Numbers k such that 9*R_k - 8 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

+20
16

3, 5, 7, 33, 45, 105, 197, 199, 281, 301, 317, 1107, 1657, 3395, 35925, 37597, 64305, 80139, 221631

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also numbers k such that 10^k - 9 is a prime.

LINKS

Table of n, a(n) for n=1..19.

Makoto Kamada, Prime numbers of the form 99...991.

Index entries for primes involving repunits

FORMULA

a(n) = A056696(n) + 1.

EXAMPLE

a(2) = 5, since 10^5 - 9 = 99991, which is prime.

MATHEMATICA

Do[ If[ PrimeQ[10^n - 9], Print[n]], {n, 0, 7000}]

CROSSREFS

Cf. A002275, A093177, A056696.

KEYWORD

nonn

AUTHOR

Alonso del Arte, Jul 07 2004

EXTENSIONS

a(12) - a(14) from Robert G. Wilson v, Oct 15 2004

a(15) - a(16) from Jason Earls, Jan 07 2008

a(17) - a(19) from Alexander Gramolin, May 13 2011

Edited by Ray Chandler, Feb 26 2012

Title corrected by Robert Price, Sep 06 2014

STATUS

approved

A102380

Irregular triangle read by rows in which row n lists prime factors (with multiplicity) of the repunit (10^n - 1)/9 (A002275(n)).

+20
15

1, 11, 3, 37, 11, 101, 41, 271, 3, 7, 11, 13, 37, 239, 4649, 11, 73, 101, 137, 3, 3, 37, 333667, 11, 41, 271, 9091, 21649, 513239, 3, 7, 11, 13, 37, 101, 9901, 53, 79, 265371653, 11, 239, 4649, 909091, 3, 31, 37, 41, 271, 2906161, 11, 17, 73, 101

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

See A003020 for other links and references.

LINKS

Rows n=1..352, flattened

Makoto Kamada, Factorizations of 11...11 (repunit).

Yousuke Koide, Factorizations of Repunit Numbers

Franz Lemmermeyer, Simerka - Quadratic Forms and Factorization, arXiv:1201.0282 [math.NT], 2011. For the 19th row = [2071723, 5363222357].

Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

EXAMPLE

First rows:

1;

11;

3, 37;

11, 101;

41, 271;

3, 7, 11, 13, 37;

239, 4649;

...

MAPLE

[seq( ifactor((10^n-1)/9), n=1..20)];

CROSSREFS

Cf. A002275, A067063, A003020, A204845, A204846, A204847, A204848.

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Nov 28 2006

EXTENSIONS

First 100 rows in b-file from T. D. Noe, Feb 27 2009

Rows n=101..322 in b-file from Ray Chandler, May 01 2017

Rows n=323..352 in b-file from Max Alekseyev, Apr 26 2022

STATUS

approved

A096507

Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

+20
14

1, 2, 6, 8, 9, 11, 20, 23, 41, 63, 66, 119, 122, 149, 252, 284, 305, 592, 746, 875, 1204, 1364, 2240, 2403, 5106, 5776, 5813, 12456, 14235, 39606, 55544, 84239, 275922

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also numbers k such that (2*10^k + 1)/3 is prime.

These numbers form a near-repdigit sequence (6)w7.

All the terms from k = 2403 through 14235 correspond to primes. - Joao da Silva (zxawyh66(AT)yahoo.com), Oct 03 2005

LINKS

Table of n, a(n) for n=1..33.

Makoto Kamada, Prime numbers of the form 66...667.

Index entries for primes involving repunits

FORMULA

a(n) = A056657(n) + 1.

EXAMPLE

k = 9 gives 2000000001/3 = 666666667, which is prime.

k = 20 gives 66666666666666666667, which is prime.

MATHEMATICA

Select[Range@ 2500, PrimeQ[FromDigits@ Table[6, {#}] + 1] &] (* or *)

Select[Range@ 2500, PrimeQ[2 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jul 04 2016 *)

CROSSREFS

Cf. A002275, A056657, A093170, A096503, A096504, A096505, A096506, A096508.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jul 12 2004

EXTENSIONS

More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

39606 and 55544 from Serge Batalov, Jun 2009

84239 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012

a(33) from Kamada data by Tyler Busby, Apr 14 2024

STATUS

approved

A067063

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

+20
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

REFERENCES

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

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A002275, A004023, A020639, A102380.

Largest factor: A003020.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Jan 03 2002

EXTENSIONS

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

STATUS

approved

A272525

Convolution of nonzero repunits (A002275) with themselves.

+20
13

1, 22, 343, 4664, 58985, 713306, 8367627, 96021948, 1083676269, 12071330590, 133058984911, 1454046639232, 15775034293553, 170096021947874, 1824417009602195, 19478737997256516, 207133058984910837, 2194787379972565158, 23182441700960219479, 244170096021947873800

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Partial sums of A014925.

LINKS

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Repunit

Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100)

FORMULA

O.g.f.: 1/((1 - 10*x)^2*(1 - x)^2).

E.g.f.: (29 + 9*x + 700*exp(9*x) + 9000*x*exp(9*x))*exp(x)/729.

a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4).

a(n) = (9*n(10^(n+2) + 1) + 7*10^(n+2) + 29)/729.

A010879(a(n)) = A010879(n+1).

MATHEMATICA

LinearRecurrence[{22, -141, 220, -100}, {1, 22, 343, 4664}, 20]

Table[(9 n (10^(n + 2) + 1) + 7 10^(n + 2) + 29)/729, {n, 0, 19}]

PROG

(PARI) A272525(n)=(9*n+7)*(10^(n+2)+1)\729+1 \\ M. F. Hasler, Nov 02 2016

CROSSREFS

Cf. A002275, A010879, A014925, A083449.

KEYWORD

base,nonn,easy

AUTHOR

Ilya Gutkovskiy, May 02 2016

STATUS

approved

A100706

Bisection of A002275.

+20
11

1, 111, 11111, 1111111, 111111111, 11111111111, 1111111111111, 111111111111111, 11111111111111111, 1111111111111111111, 111111111111111111111, 11111111111111111111111

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Also the binary representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell for Rules 151, 159, 183, 191, 215, 222, 223, 247, 254 and 255. - Robert Price, Feb 21 2016

The aerated sequence 1, 0, 111, 0, 11111, 0, 1111111, ... is a linear divisibility sequence of order 4. It is the case P1 = 0, P2 = -9^2, Q = -10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A007583, A095372 and A299960. - Peter Bala, Aug 28 2019

REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

LINKS

Table of n, a(n) for n=0..11.

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

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.

S. Wolfram, A New Kind of Science

Index to Elementary Cellular Automata

Index entries for sequences related to cellular automata

Index entries for linear recurrences with constant coefficients, signature (101,-100).

FORMULA

Numbers composed entirely of 2*n+1 concatenated 1's for n >= 0.

O.g.f.: (1+10*x)/((-1+x)*(-1+100*x)). - R. J. Mathar, Apr 03 2008

From Klaus Purath, Sep 23 2020: (Start)

a(n) = Sum_{i = 0..2*n} 10^i.

a(n) = 101*a(n-1) - 100*a(n-2).

a(n) = 110*10^(2*n-2) + a(n-1).

a(n) = 100*a(n-1) + 11.