A000042 -id:A000042 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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.

David Wasserman, Table of n, a(n) for n = 0..1000
Eudes Antonio Costa and Fernando Soares Carvalho, On repunit polynomials sequence, Braz. Elec. J. Math. (2024). See pp. 2, 15.
Eudes Antonio Costa, Douglas Catulio Santos, Paula Maria Machado Cruz Catarino, and Elen Viviani Pereira Spreafico, On Gaussian and Quaternion Repunit Numbers, Rev. Mat. UFOP (Brazil, 2024) Vol. 2. See p. 2.
Eudes Antonio Costa, Paula Maria Machado Cruz Catarino, and Douglas Catulio Santos, A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence, Symmetry (2024) Vol. 17, No. 1, 28.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Douglas Catulio Santos, Eudes Antonio Costa, and Paula Maria Machado Cruz Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
W. M. Snyder, Factoring Repunits, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.
Amelia Carolina Sparavigna, On Repunits, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Repunit.
Eric Weisstein's World of Mathematics, Demlo Number.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Wikipedia, Repunit.
Amin Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6.
Stephen Wolfram, A New Kind of Science.
Samuel Yates, The Mystique of Repunits, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.
Index to Elementary Cellular Automata.
Index entries for 10-automatic sequences.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (11,-10).
Index entries for "core" sequences.

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.

a002275 = (`div` 9) . subtract 1 . (10 ^)
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

[(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014

seq((10^k - 1)/9, k=0..30); # Wesley Ivan Hurt, Sep 28 2013

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 *)

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 */

a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009

my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016

print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022

[lucas_number1(n, 11, 10) for n in range(21)]  # Zerinvary Lajos, Apr 27 2009

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)

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

List of binary numbers. (This comment is to assist people searching for that particular phrase. - N. J. A. Sloane, Apr 08 2016)
Or, numbers that are sums of distinct powers of 10.
Or, numbers having only digits 0 and 1 in their decimal representation.
Complement of A136399; A064770(a(n)) = a(n). - Reinhard Zumkeller, Dec 30 2007
From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 1.
Thus nonnegative integers n in base 10 such that kn can be calculated by normal addition (i.e., n + n + ... + n, with k n's (but not necessarily k + k + ... + k, with n k's)) or multiplication without requiring any carry operations for 0 <= k <= 9. (End)
For n > 1: A257773(a(n)) = 10, numbers that are Belgian-k for k=0..9. - Reinhard Zumkeller, May 08 2015
For any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
N is in this sequence iff A007953(N) = A101337(N). A028897 is a left inverse. - M. F. Hasler, Nov 18 2019
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023

			a(6)=110 because (1/2)*((1-(-1)^6)*10^0 + (1-(-1)^3)*10^1 + (1-(-1)^1)*10^2) = 10 + 100.
G.f. = x + 10*x^2 + 11*x^3 + 100*x^4 + 101*x^5 + 110*x^6 + 111*x^7 + 1000*x^8 + ...
.
  000    The numbers < 2^n can be regarded as vectors with
  001    a fixed length n if padded with zeros on the left
  010    side. This represents the n-fold Cartesian product
  011    over the set {0, 1}. In the example on the left,
  100    n = 3. (See also the second Python program.)
  101    Binary vectors in this format can also be seen as a
  110    representation of the subsets of a set with n elements.
  111    - _Peter Luschny_, Jan 22 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 21.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991, p. 383.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..32768 (first 8192 terms from Franklin T. Adams-Watters)
Heinz Gumin, Herrn von Leibniz' Rechnung mit Null und Eins, Siemens AG, 3. Auflage 1979 -- contains facsimiles of Leibniz's papers from 1679 and 1703.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
G. W. Leibniz, Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 & 1; avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de l'Académie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).
N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.
Index entries for sequences related to binary expansion of n.

The basic sequences concerning the binary expansion of n are this one, A000120 (Hammingweight: sum of bits), A000788 (partial sums of A000120), A000069 (A000120 is odd), A001969 (A000120 is even), A023416 (number of bits 0), A059015 (partial sums). Bisections A099820 and A099821.
Cf. A028897 (convert binary to decimal).
Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.
Cf. A000290, A001737, A007953, A030308, A054055, A064770, A101337, A136399.

a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n',m) = divMod n 2
-- Reinhard Zumkeller, Jan 10 2012

A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009

Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
Table[Sum[ (Floor[( Mod[f/2 ^n, 2])])*(10^n) , {n, 0, Floor[Log[2, f]]}], {f, 1, 100}] (* José de Jesús Camacho Medina, Jul 24 2014 *)
FromDigits/@Tuples[{1,0},6]//Sort (* Harvey P. Dale, Aug 10 2017 *)

{a(n) = subst( Pol( binary(n)), x, 10)}; /* Michael Somos, Jun 07 2002 */

{a(n) = if( n<=0, 0, n%2 + 10*a(n\2))}; /* Michael Somos, Jun 07 2002 */

a(n)=fromdigits(binary(n),10) \\ Charles R Greathouse IV, Apr 08 2015

def a(n): return int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021

from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
# Peter Luschny, Jan 22 2024

a(n) = Sum_{i=0..m} d(i)*10^i, where Sum_{i=0..m} d(i)*2^i is the base 2 representation of n.
a(n) = (1/2)*Sum_{i>=0} (1-(-1)^floor(n/2^i))*10^i. - Benoit Cloitre, Nov 20 2001
a(n) = A097256(n)/9.
a(2n) = 10*a(n), a(2n+1) = a(2n)+1.
G.f.: 1/(1-x) * Sum_{k>=0} 10^k * x^(2^k)/(1+x^(2^k)) - for sequence as decimal integers. - Franklin T. Adams-Watters, Jun 16 2006
a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009
a(n) = Sum_{k>=0} A030308(n,k)*10^k. - Philippe Deléham, Oct 19 2011
For n > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012
a(n) = Sum_{k=0..floor(log_2(n))} floor((Mod(n/2^k, 2)))*(10^k). - José de Jesús Camacho Medina, Jul 24 2014

A007089 Numbers in base 3.

Original entry on oeis.org

0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 222, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1200, 1201, 1202, 1210, 1211

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Nonnegative integers with no decimal digit > 2. Thus nonnegative integers in base 10 whose quadrupling by normal addition or multiplication requires no carry operation. - Rick L. Shepherd, Jun 25 2009

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.3 Positional Notation, p. 47.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
The Unicode Consortium, Tai Xuan Jing Symbols.
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.

Cf. A000042, A007088, A007090, A007091, A007092, A007093, A007094, A007095, A077267, A062756, A081603, A081604, A054635, A003137.

Primes when read as if in base 10: A036954.

a007089 0 = 0
a007089 n = 10 * a007089 n' + m where (n', m) = divMod n 3
-- Reinhard Zumkeller, Feb 19 2012

A007089 := proc(n) option remember;
if n <= 0 then 0
else
  if (n mod 3) = 0 then 10*procname(n/3) else procname(n-1) + 1 fi
fi end:
[seq(A007089(n), n=0..729)]; # - N. J. A. Sloane, Mar 09 2019

Table[ FromDigits[ IntegerDigits[n, 3]], {n, 0, 50}]

a(n)=if(n<1,0,if(n%3,a(n-1)+1,10*a(n/3)))

a(n)=fromdigits(digits(n,3)) \\ Charles R Greathouse IV, Jan 08 2017

def A007089(n):
  n,s = divmod(n,3); t = 1
  while n: n,r = divmod(n,3); t *= 10; s += r*t
  return s # M. F. Hasler, Feb 15 2023

a(0)=0, a(n) = 10*a(n/3) if n==0 (mod 3), a(n) = a(n-1) + 1 otherwise. - Benoit Cloitre, Dec 22 2002

a(n) = 10*a(floor(n/3)) + (n mod 3) if n > 0, a(0) = 0. - M. F. Hasler, Feb 15 2023

More terms from James Sellers, May 01 2000

A007090 Numbers in base 4.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, 200, 201, 202, 203, 210, 211, 212, 213, 220, 221, 222, 223, 230, 231, 232, 233, 300, 301, 302, 303, 310, 311, 312, 313, 320, 321, 322, 323, 330, 331, 332, 333

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Nonnegative integers with no decimal digit > 3. Thus nonnegative integers in base 10 whose tripling (trebling) by normal addition or multiplication requires no carry operation. - Rick L. Shepherd, Jun 25 2009

Interpreted in base 10: a(x)+a(y) = a(z) => x+y = z. The converse is not true in general. - Karol Bacik, Sep 27 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A007608, A000042, A007088 (base 2), A007089 (base 3), A007091 (base 5), A007092 (base 6), A007093 (base 7), A007094 (base 8), A007095 (base 9), A193890, A107715.

a007090 0 = 0
a007090 n = 10 * a007090 n' + m where (n', m) = divMod n 4
-- Reinhard Zumkeller, Apr 08 2013, Aug 11 2011

A007090 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,4): return op(convert(l,base,10,10^nops(l))): end: seq(A007090(n),n=0..54); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 60}]

a(n)=if(n<1,0,if(n%4,a(n-1)+1,10*a(n/4)))

A007090(n)=sum(i=1,#n=digits(n,4),n[i]*10^(#n-i)) \\ M. F. Hasler, Jul 25 2015 (Corrected by Jinyuan Wang, Oct 02 2019)

apply( A007090(n)=fromdigits(digits(n,4)), [0..66]) \\ M. F. Hasler, Nov 18 2019

a(n) = Sum_{d(i)*10^i: i=0, 1, ..., m}, where Sum_{d(i)*4^i: i=0, 1, ..., m} is the base 4 representation of n.

a(0) = 0, a(n) = 10*a(n/4) if n==0 (mod 4), a(n) = a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

A007091 Numbers in base 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 200, 201, 202, 203, 204, 210, 211, 212, 213, 214, 220, 221, 222, 223, 224, 230

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 4.
Thus nonnegative integers in base 10 whose doubling by normal addition or multiplication requires no carry operation. (End)
It appears that this sequence corresponds to the numbers n for which twice the sum of digits of n is the sum of digits of 2*n. - Rémy Sigrist, Nov 22 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007092 (base 6), A007093 (base 7), A007094 (base 8), A007095 (base 9).

Cf. A037454, A037462, A102491.

A007091 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,5): return op(convert(l,base,10,10^nops(l))): end: seq(A007091(n),n=0..58); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 5]], {n, 0, 60}]

a(n)=if(n<1,0,if(n%5,a(n-1)+1,10*a(n/5)))

apply( A007091(n)=fromdigits(digits(n,5)), [0..66]) \\ M. F. Hasler, Nov 18 2019

from gmpy2 import digits
def A007091(n): return int(digits(n,5)) # Chai Wah Wu, Dec 26 2021

a(0)=0 a(n)=10*a(n/5) if n==0 (mod 5) a(n)=a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

a(n) = n + 1/2*Sum_{k >= 1} 10^k*floor(n/5^k). Cf. A037454, A037462 and A102491. - Peter Bala, Dec 01 2016

A007095 Numbers in base 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Also numbers without 9 as a digit.

Complement of A011539: A102683(a(n)) = 0; A068505(a(n)) != a(n)). - Reinhard Zumkeller, Dec 29 2011

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007091 (base 5), A007092 (base 6), A007093 (base 7), A007094 (base 8); A057104, A037479.
Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8).
Cf. A031076, A031087.
Cf. A082838.

a007095 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + r   where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014, Dec 29 2011

[ n: n in [0..74] | not 9 in Intseq(n) ];  // Bruno Berselli, May 28 2011

A007095 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,9): return op(convert(l,base,10,10^nops(l))): end: seq(A007095(n),n=0..67); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 9]], {n, 0, 75}]

a(n)=if(n<1,0,if(n%9,a(n-1)+1,10*a(n/9)))

A007095(n)=fromdigits(digits(n, 9)) \\ Michel Marcus, Dec 29 2018

# and others: see OEIS Wiki page (cf. LINKS).

from gmpy2 import digits
def A007095(n): return int(digits(n,9)) # Chai Wah Wu, May 06 2025

seq 0 1000 | grep -v 9; # Joerg Arndt, May 29 2011

a(0) = 0, a(n) = 10*a(n/9) if n==0 (mod 9), a(n) = a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

Sum_{n>1} 1/a(n) = A082838 = 22.92067... (Kempner series). - Bernard Schott, Dec 29 2018; edited by M. F. Hasler, Jan 13 2020

A007094 Numbers in base 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 100, 101, 102, 103, 104, 105, 106, 107, 110, 111

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Wikipedia, Octal.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A057104; A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007091 (base 5), A007092 (base 6), A007093 (base 7), A007095 (base 9).

a007094 0 = 0
a007094 n = 10 * a007094 n' + m where (n', m) = divMod n 8
-- Reinhard Zumkeller, Aug 29 2013

A007094 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,8): return op(convert(l,base,10,10^nops(l))): end: seq(A007094(n), n=0..66); # Nathaniel Johnston, May 06 2011

Table[FromDigits[IntegerDigits[n, 8]], {n, 0, 70}]

a(n)=if(n<1,0,if(n%8,a(n-1)+1,10*a(n/8)))

apply( A007094(n)=fromdigits(digits(n,8)), [0..77]) \\ M. F. Hasler, Nov 18 2019

def a(n): return int(oct(n)[2:])
print([a(n) for n in range(74)]) # Michael S. Branicky, Jun 28 2021

a(0) = 0; a(n) = 10*a(n/8) if n == 0 (mod 8); a(n) = a(n-1) + 1 otherwise. - Benoit Cloitre, Dec 22 2002

G.f.: sum(d>=0, 10^d*(x^(8^d) +2*x^(2*8^d) +3*x^(3*8^d) +4*x^(4*8^d) +5*x^(5*8^d) +6*x^(6*8^d) +7*x^(7*8^d)) * (1-x^(8^d)) / ((1-x^(8^(d+1)))*(1-x))). - Robert Israel, Aug 03 2014

cp's OEIS Frontend

A077488 All possible products in increasing order of two distinct members of the unary sequence 1, 11, 111, 1111, 11111, ...given by A000042.

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A086565 Smallest k such that (10^k - 1)/9 has n distinct prime divisors. Or a(n)= smallest value of k such that A000042(k) has exactly n distinct prime divisors.

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A083826 Smallest k such that k+ 9*(A000042(n-1)*{n, mod 10) is digit reversal of k. A000042(n) = (10^n-1)/9.,a(10n) = 0.

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

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A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

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A007089 Numbers in base 3.

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A007090 Numbers in base 4.

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A083826 Smallest k such that k+ 9(A000042(n-1){n, mod 10) is digit reversal of k. A000042(n) = (10^n-1)/9.,a(10n) = 0.