cp's OEIS Frontend

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

%I A002275 #300 Aug 04 2025 01:10:17
%S A002275 0,1,11,111,1111,11111,111111,1111111,11111111,111111111,1111111111,
%T A002275 11111111111,111111111111,1111111111111,11111111111111,
%U A002275 111111111111111,1111111111111111,11111111111111111,111111111111111111,1111111111111111111,11111111111111111111
%N A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.
%C A002275 R_n is a string of n 1's.
%C A002275 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
%C A002275 Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - _Zak Seidov_, Dec 05 2008
%C A002275 For n >= 0: a(n) = (A000225(n) written in base 2). - _Jaroslav Krizek_, Jul 27 2009, edited by _M. F. Hasler_, Jul 03 2020
%C A002275 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
%C A002275 Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - _Bernard Schott_, Dec 24 2012
%C A002275 Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - _Bernard Schott_, Dec 24 2012
%C A002275 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
%C A002275 For n>=2 the multiplicative order of 10 modulo the a(n) is n. - _Robert G. Wilson v_, Aug 20 2014
%C A002275 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
%C A002275 From _Peter Bala_, Sep 20 2015: (Start)
%C A002275 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)
%C A002275 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
%C A002275 If p > 5 is a prime, then p divides a(p-1). - _Thomas Ordowski_, Apr 10 2016
%C A002275 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
%C A002275 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
%C A002275 The term "repunit" was coined by Albert H. Beiler in 1964. - _Amiram Eldar_, Nov 13 2020
%C A002275 q-integers for q = 10. - _John Keith_, Apr 12 2021
%C A002275 Binomial transform of A001019 with leading zero. - _Jules Beauchamp_, Jan 04 2022
%D A002275 Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.
%D A002275 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 235-237.
%D A002275 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.
%D A002275 Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.
%D A002275 Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.
%H A002275 David Wasserman, <a href="/A002275/b002275.txt">Table of n, a(n) for n = 0..1000</a>
%H A002275 Eudes Antonio Costa and Fernando Soares Carvalho, <a href="https://doi.org/10.14393/BEJOM-v5-2024-72280">On repunit polynomials sequence</a>, Braz. Elec. J. Math. (2024). See pp. 2, 15.
%H A002275 Eudes Antonio Costa, Douglas Catulio Santos, Paula Maria Machado Cruz Catarino, and Elen Viviani Pereira Spreafico, <a href="https://doi.org/10.5281/zenodo.14563273">On Gaussian and Quaternion Repunit Numbers</a>, Rev. Mat. UFOP (Brazil, 2024) Vol. 2. See p. 2.
%H A002275 Eudes Antonio Costa, Paula Maria Machado Cruz Catarino, and Douglas Catulio Santos, <a href="https://doi.org/10.3390/sym17010028">A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence</a>, Symmetry (2024) Vol. 17, No. 1, 28.
%H A002275 Dmytro S. Inosov and Emil Vlasák, <a href="https://arxiv.org/abs/2410.21427">Cryptarithmically unique terms in integer sequences</a>, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.
%H A002275 Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.
%H A002275 Douglas Catulio Santos, Eudes Antonio Costa, and Paula Maria Machado Cruz Catarino, <a href="https://doi.org/10.3390/axioms14030203">On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence</a>, Axioms 14, 203, (2025). See p. 4.
%H A002275 W. M. Snyder, <a href="http://www.jstor.org/stable/2321382">Factoring Repunits</a>, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.
%H A002275 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.2639620">On Repunits</a>, Politecnico di Torino (Italy, 2019).
%H A002275 Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2044">Composition Operations of Generalized Entropies Applied to the Study of Numbers</a>, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
%H A002275 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H A002275 Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences (2019) Vol. 8, No. 10.
%H A002275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a>.
%H A002275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DemloNumber.html">Demlo Number</a>.
%H A002275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.
%H A002275 Wikipedia, <a href="http://en.wikipedia.org/wiki/Repunit">Repunit</a>.
%H A002275 Amin Witno, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Witno/witno6.html">A Family of Sequences Generating Smith Numbers</a>, J. Int. Seq. 16 (2013) #13.4.6.
%H A002275 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>.
%H A002275 Samuel Yates, <a href="http://www.jstor.org/stable/2689643">The Mystique of Repunits</a>, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.
%H A002275 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>.
%H A002275 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%H A002275 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.
%H A002275 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%H A002275 <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%F A002275 a(n) = 10*a(n-1) + 1, a(0)=0.
%F A002275 a(n) = A000042(n) for n >= 1.
%F A002275 Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - _Paul Barry_, Mar 24 2004
%F A002275 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
%F A002275 a(n) = 11*a(n-1) - 10*a(n-2), a(0)=0, a(1)=1. - _Lekraj Beedassy_, Jun 07 2006
%F A002275 a(n) = A125118(n,9) for n>8. - _Reinhard Zumkeller_, Nov 21 2006
%F A002275 a(n) = A075412(n)/A002283(n). - _Reinhard Zumkeller_, May 31 2010
%F A002275 a(n) = a(n-1) + 10^(n-1) with a(0)=0. - _Vincenzo Librandi_, Jul 22 2010
%F A002275 a(n) = A242614(n,A242622(n)). - _Reinhard Zumkeller_, Jul 17 2014
%F A002275 E.g.f.: (exp(9*x) - 1)*exp(x)/9. - _Ilya Gutkovskiy_, May 11 2016
%F A002275 a(n) = Sum_{k=0..n-1} 10^k. - _Torlach Rush_, Nov 03 2020
%F A002275 Sum_{n>=1} 1/a(n) = A065444. - _Amiram Eldar_, Nov 13 2020
%F A002275 From _Elmo R. Oliveira_, Aug 02 2025: (Start)
%F A002275 a(n) = A002283(n)/9 = A105279(n)/10.
%F A002275 a(n) = A010785(A017173(n-1)) for n >= 1. (End)
%p A002275 seq((10^k - 1)/9, k=0..30); # _Wesley Ivan Hurt_, Sep 28 2013
%t A002275 Table[(10^n - 1)/9, {n, 0, 19}] (* _Alonso del Arte_, Nov 15 2011 *)
%t A002275 Join[{0},Table[FromDigits[PadRight[{},n,1]],{n,20}]] (* _Harvey P. Dale_, Mar 04 2012 *)
%o A002275 (PARI) a(n)=(10^n-1)/9; \\ _Michael B. Porter_, Oct 26 2009
%o A002275 (PARI) my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ _Altug Alkan_, Apr 10 2016
%o A002275 (Sage) [lucas_number1(n, 11, 10) for n in range(21)]  # _Zerinvary Lajos_, Apr 27 2009
%o A002275 (Haskell)
%o A002275 a002275 = (`div` 9) . subtract 1 . (10 ^)
%o A002275 a002275_list = iterate ((+ 1) . (* 10)) 0
%o A002275 -- _Reinhard Zumkeller_, Jul 05 2013, Feb 05 2012
%o A002275 (Maxima)
%o A002275 a[0]:0$
%o A002275 a[1]:1$
%o A002275 a[n]:=11*a[n-1]-10*a[n-2]$
%o A002275 A002275(n):=a[n]$
%o A002275 makelist(A002275(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A002275 (Magma) [(10^n-1)/9: n in [0..25]]; // _Vincenzo Librandi_, Nov 06 2014
%o A002275 (Python)
%o A002275 print([(10**n-1)//9 for n in range(100)]) # _Michael S. Branicky_, Apr 30 2022
%Y A002275 Partial sums of 10^n (A011557). Factors: A003020, A067063.
%Y A002275 Bisections give A099814, A100706.
%Y A002275 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.
%Y A002275 Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.
%Y A002275 Cf. A000225, A001019, A001045, A002450, A004215, A015565, A125118, A242614, A242622.
%Y A002275 Cf. A010785, A017173, A105279.
%K A002275 easy,nonn,nice,core
%O A002275 0,3
%A A002275 _N. J. A. Sloane_