cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

Original entry on oeis.org

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

R_n is a string of n 1's.
Base-4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 1. - Paul Barry, Mar 12 2004
Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - Zak Seidov, Dec 05 2008
For n >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012
Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012
The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013
For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014
The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014
From Peter Bala, Sep 20 2015: (Start)
Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)
Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016
If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016
0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016
Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017
The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020
q-integers for q = 10. - John Keith, Apr 12 2021
Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

References

  • Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 235-237.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.
  • Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.
  • Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.

Links

Crossrefs

Partial sums of 10^n (A011557). Factors: A003020, A067063.
Bisections give A099814, A100706.
Cf. A000042, A002276, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A004023, A046053, A059988, A065444, A075412, A075415, A083278, A095370, A102380, A125134, A178635, A204845, A204846, A204847, A204848, A206244.
Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.
Cf. A000225, A001019, A001045, A002450, A004215, A015565, A125118, A242614, A242622.
Cf. A010785, A017173, A105279.

Programs

  • Haskell
    a002275 = (`div` 9) . subtract 1 . (10 ^)
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012
  • Magma
    [(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014
  • Maple
    seq((10^k - 1)/9, k=0..30); # Wesley Ivan Hurt, Sep 28 2013
  • Mathematica
    Table[(10^n - 1)/9, {n, 0, 19}] (* Alonso del Arte, Nov 15 2011 *)
Join[{0},Table[FromDigits[PadRight[{},n,1]],{n,20}]] (* Harvey P. Dale, Mar 04 2012 *)
  • Maxima
    a[0]:0$
a[1]:1$
a[n]:=11*a[n-1]-10*a[n-2]$
A002275(n):=a[n]$
makelist(A002275(n),n,0,30); /* Martin Ettl, Nov 05 2012 */
  • PARI
    a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009
  • PARI
    my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016
  • Python
    print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022
  • Sage
    [lucas_number1(n, 11, 10) for n in range(21)]  # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 10*a(n-1) + 1, a(0)=0.
a(n) = A000042(n) for n >= 1.
Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - Paul Barry, Mar 24 2004
G.f.: x/((1-10*x)*(1-x)). Regarded as base b numbers, g.f. x/((1-b*x)*(1-x)). - Franklin T. Adams-Watters, Jun 15 2006
a(n) = 11*a(n-1) - 10*a(n-2), a(0)=0, a(1)=1. - Lekraj Beedassy, Jun 07 2006
a(n) = A125118(n,9) for n>8. - Reinhard Zumkeller, Nov 21 2006
a(n) = A075412(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = A242614(n,A242622(n)). - Reinhard Zumkeller, Jul 17 2014
E.g.f.: (exp(9*x) - 1)*exp(x)/9. - Ilya Gutkovskiy, May 11 2016
a(n) = Sum_{k=0..n-1} 10^k. - Torlach Rush, Nov 03 2020
Sum_{n>=1} 1/a(n) = A065444. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Aug 02 2025: (Start)
a(n) = A002283(n)/9 = A105279(n)/10.
a(n) = A010785(A017173(n-1)) for n >= 1. (End)

A102380 Irregular triangle read by rows in which row n lists prime factors (with multiplicity) of the repunit (10^n - 1)/9 (A002275(n)).

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane, Nov 28 2006

Keywords

Comments

See A003020 for other links and references.

Examples

			First rows:
    1;
   11;
    3,   37;
   11,  101;
   41,  271;
    3,    7, 11, 13, 37;
  239, 4649;
  ...

Links

Crossrefs

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

Programs

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

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

A204845 Irregular triangle read by rows in which row n lists primitive prime factors of the repunit (10^n - 1)/9 (A002275(n)).

Original entry on oeis.org

1, 11, 3, 37, 101, 41, 271, 7, 13, 239, 4649, 73, 137, 333667, 9091, 21649, 513239, 9901, 53, 79, 265371653, 909091, 31, 2906161, 17, 5882353, 2071723, 5363222357, 19, 52579, 1111111111111111111, 3541, 27961, 43, 1933, 10838689, 23, 4093, 8779, 11111111111111111111111
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2012

Keywords

Examples

			Triangle begins:
1
11
3 37
101
41 271
7 13
239 4649
73 137
333667
9091
...

Links

Crossrefs

Cf. A002275, A102380, A204846, A204847, A204848.

Programs

  • Maple
    with(numtheory):
S:= proc(n) option remember;
      `if`(n=1, {1}, S(n-1) union factorset ((10^n-1)/9))
    end:
T:= n-> sort([(S(n) minus `if`(n=1, {}, S(n-1)))[]])[]:
seq(T(n), n=1..30);  # Alois P. Heinz, Feb 17 2012
  • Mathematica
    S[n_] := S[n] = If[n==1, {1}, S[n-1] ~Union~ FactorInteger[(10^n-1)/9][[ All, 1]]]; T[n_] := Sort[S[n] ~Complement~ If[n==1, {}, S[n-1]]]; Table[ T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 17 2012

A204847 Primitive cofactor of n-th repunit A002275(n).

Original entry on oeis.org

1, 11, 111, 101, 11111, 91, 1111111, 10001, 333667, 9091, 11111111111, 9901, 1111111111111, 909091, 90090991, 100000001, 11111111111111111, 999001, 1111111111111111111, 99009901, 900900990991, 826446281, 11111111111111111111111, 99990001, 100001000010000100001
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2012

Keywords

Comments

Except for a(1) = 1 and a(3) = 111, this is the Zsigmondy numbers for a = 10, b = 1: Zs(n, 10, 1) is the greatest divisor of 10^n - 1^n that is coprime to 10^m - 1^m for all positive integers m < n. The prime terms are called unique primes or unique period primes (A007615).
Differs from A019328 for n = 1, 9, 22, 27, 42, ... - Jianing Song, Apr 30 2018

Links

Crossrefs

Cf. A002275, A019328, A102380, A204845, A204846.

Programs

  • PARI
    lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), f = factor((10^n-1)/9)[,1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k]));); print1(prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf);););} \\ Michel Marcus, May 18 2018

Formula

Equals A002275(n)/(product of terms in n-th row of A204845).

Extensions

a(11)-a(24) from Jianing Song, Apr 30 2018
a(25) from Jinyuan Wang, May 02 2021

A204848 Algebraic cofactor of n-th repunit A002275(n).

Original entry on oeis.org

1, 1, 1, 11, 1, 1221, 1, 1111, 333, 122221, 1, 11222211, 1, 12222221, 1233321, 11111111, 1, 111222222111, 1, 112222222211, 123333321, 1344444444431, 1, 1111222222221111, 11111, 12222222222221, 333333333, 1122222222222211, 1, 1011121222222221211101, 1, 1111111111111111, 1233333333321
Offset: 1

Views

Author

N. J. A. Sloane, Jan 19 2012

Keywords

Links

Crossrefs

Cf. A002275, A102380, A204845, A204846, A204847.

Programs

  • PARI
    lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), rn = (10^n-1)/9; f = factor(rn)[, 1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k])); ); print1(rn/prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf); ); ); } \\ Michel Marcus, May 20 2018

Formula

Equals A002275(n)/(product of terms in n-th row of A204846).

Extensions

More terms from Michel Marcus, May 20 2018
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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