A000042 -id:A000042 - cp's OEIS frontend

A065444 Decimal expansion of 9*Sum_{k>=1} 1/(10^k-1).

1, 1, 0, 0, 9, 1, 8, 1, 9, 0, 8, 3, 6, 2, 0, 0, 7, 3, 6, 3, 7, 9, 8, 5, 5, 1, 0, 1, 6, 5, 4, 3, 8, 0, 0, 4, 3, 2, 0, 3, 4, 5, 4, 3, 9, 7, 8, 7, 3, 2, 8, 1, 6, 5, 6, 3, 5, 9, 8, 9, 0, 2, 2, 0, 7, 3, 4, 3, 8, 3, 4, 9, 0, 2, 1, 9, 8, 3, 4, 7, 4, 8, 8, 9, 2, 0, 0, 3, 4, 9, 2, 1, 8, 0, 0, 7, 0, 4, 0, 2, 3, 5

Offset: 1

N. J. A. Sloane, Nov 18 2001

This constant is the infinite sum of the reciprocals of repunits, R_n, with n>0 (A002275). - Enrique Pérez Herrero, Dec 06 2009

			1.10091819083620073637985510165438004320345439787328165635989...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Equals 9*A073668.

Cf. A000042, A002275, A067617 (continued fraction).

RealDigits[9*N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]
A065444=RealDigits[ Block[{$MaxExtraPrecision = 100}, N[9*Sum[(-1 + 10^i)^-1, {i, 1, Infinity}], 130]]][[1]] (* Enrique Pérez Herrero, Dec 06 2009 *)
First[RealDigits[9 (Log10[10/9] - QPolyGamma[0, 1, 1/10]/Log[10]), 10, 120]] (* Jan Mangaldan, Apr 25 2016 *)

{ default(realprecision, 2080); x=9*suminf(k=1, 1/(10^k - 1)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065444.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Equals 9 * Sum_{k>=1} (1+x^k)/(1-x^k) * x^(k^2) where x = 1/10. This allows fast computation for this and similar sequences (involving Sum_{k>=1} x^k/(1-x^k) for some x < 1 ). - Joerg Arndt, Apr 25 2016

More terms from John W. Layman, Nov 19 2001

...733 (50th digit) expanded to ...7328165 etc. by Frank Ellermann, Feb 23 2002

A086066 a(n) = Sum_{d in D(n)} 2^d, where D(n) = set of digits of n in decimal representation.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 3, 2, 6, 10, 18, 34, 66, 130, 258, 514, 5, 6, 4, 12, 20, 36, 68, 132, 260, 516, 9, 10, 12, 8, 24, 40, 72, 136, 264, 520, 17, 18, 20, 24, 16, 48, 80, 144, 272, 528, 33, 34, 36, 40, 48, 32, 96, 160, 288, 544, 65, 66, 68, 72, 80

Offset: 0

Reinhard Zumkeller, Jul 08 2003

For bitwise logical operations AND and OR:
a(m) = (a(m) AND a(n)) iff D(m) is a subset of D(n),
(a(m) AND a(n)) = 0 iff D(m) and D(n) are disjoint,
a(m) = (a(m) OR a(n)) iff D(n) is a subset of D(m),
a(m) = a(n) iff D(m) = D(n);
A086067(n) = A007088(a(n)).
From Reinhard Zumkeller, Sep 18 2009: (Start)
a(A052382(n)) mod 2 = 0; a(A011540(n)) mod 2 = 1;
for n > 0: a(A000004(n))=1, a(A000042(n))=2, a(A011557(n))=3, a(A002276(n))=4, a(A111066(n))=6, a(A002277(n))=8, a(A002278(n))=16, a(A002279(n))=32, a(A002280(n))=64, a(A002281(n))=128, a(A002282(n))=256, a(A002283(n))=512;
a(n) <= 1023. (End)

			n=242, D(242) = {2,4}: a(242) = 2^2 + 2^4 = 20.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

A086066 := proc(n) local d: if(n=0)then return 1: fi: d:=convert(convert(n,base,10),set): return add(2^d[j],j=1..nops(d)): end: seq(A086066(n),n=0..64); # Nathaniel Johnston, May 31 2011

A255805 Numbers with no zeros in base-8 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A047592, A207481.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Octal
Index entries for 2-automatic sequences.

Cf. A007094, A100970 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255808 (base 9), A052382 (base 10).

a255805 n = a255805_list !! (n-1)
a255805_list = iterate f 1 where
   f x = 1 + if r < 7 then x else 8 * f x'  where (x', r) = divMod x 8

Select[Range[100],DigitCount[#,8,0]==0&] (* Harvey P. Dale, Jun 08 2015 *)

isok(m) = vecmin(digits(m,8)) > 0; \\ Michel Marcus, Jan 23 2022

def ok(n): return '0' not in oct(n)[2:]
print([k for k in range(85) if ok(k)]) # Michael S. Branicky, Jan 23 2022

from sympy import integer_log
def A255805(n):
    m = integer_log(k:=6*n+1,7)[0]
    return sum(1+(k-7**m)//(6*7**j)%7<<3*j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A248910 Numbers with no zeros in base-6 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A039215, A047253, A184522, A187390, A194386.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Senary
Index entries for 6-automatic sequences.

Cf. A007092, A100969 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A255805 (base 8), A255808 (base 9), A052382 (base 10).

a248910 n = a248910_list !! (n-1)
a248910_list = iterate f 1 where
   f x = 1 + if r < 5 then x else 6 * f x'  where (x', r) = divMod x 6

Select[Range[100], DigitCount[#,6, 0] == 0 &] (* Paolo Xausa, Jun 29 2025 *)

isok(m) = vecmin(digits(m, 6)) > 0; \\ Michel Marcus, Jan 23 2022

from sympy import integer_log
def A248910(n):
    m = integer_log(k:=(n<<2)+1,5)[0]
    return sum((1+(k-5**m)//(5**j<<2)%5)*6**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A255808 Numbers with no zeros in base-9 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Mar 08 2015

a(n) = A168183(n) for n <= 72.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 3-automatic sequences.

Cf. A007095, A100973 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255805 (base 8), A052382 (base 10).

a255808 n = a255808_list !! (n-1)
a255808_list = iterate f 1 where
   f x = 1 + if r < 8 then x else 9 * f x'  where (x', r) = divMod x 9

Select[Range[100],DigitCount[#,9,0]==0&] (* or *) With[{upto=100}, Complement[ Range[upto],9*Range[Floor[upto/9]]]] (* Harvey P. Dale, May 29 2019 *)

isok(n) = vecmin(digits(n, 9)) != 0; \\ Michel Marcus, Jun 29 2019

def A255808(n):
    m = ((k:=7*n+1).bit_length()-1)//3
    return sum((1+((k-(1<<3*m))//(7<<3*j)&7))*9**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A322761 Irregular triangle read by rows in which n-th row lists all partitions of n, in graded reverse lexicographic ordering, using a compressed notation.

Original entry on oeis.org

1, 2, 11, 3, 21, 111, 4, 31, 22, 211, 1111, 5, 41, 32, 311, 221, 2111, 11111, 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111, 7, 61, 52, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 22111, 211111, 1111111

Offset: 1

N. J. A. Sloane, Dec 30 2018

Officially this is deprecated, since one cannot distinguish between (for example) parts which are 11 and parts which are 1,1. However, it is in common use and is included for completeness. See A036037, A080577, etc., for uncompressed versions.

			Triangle begins:
1,
2, 11,
3, 21, 111,
4, 31, 22, 211, 1111,
5, 41, 32, 311, 221, 2111, 11111,
6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111,
7, 61, 52, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 22111, 211111, 1111111,
...
...

Alois P. Heinz, Rows n = 1..28, flattened

Cf. A000041 (number of terms in row n), A036037, A080577.
See also A006128.
First column gives A000027.
Last elements of rows give A000042.

b:= (n, i)-> `if`(n=0 or i=1, [cat(1$n)], [map(x->
    cat(i, x), b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(parse, b(n$2))[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 30 2018

revlexsort[f_, c_] := OrderedQ[PadRight[{c, f}]];
Table[FromDigits /@ Sort[IntegerPartitions[n], revlexsort], {n, 1, 8}] // Flatten (* Jean-François Alcover, Oct 20 2020, after Gus Wiseman in A080577 *)

A071126 Length of least repunit which is a multiple of the n-th prime, or 0 if no such multiple exists.

Original entry on oeis.org

0, 3, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79, 110

Offset: 1

Lekraj Beedassy, May 28 2002

nonn

If prime(n) = p then a(n) is a divisor of p-1. - Amarnath Murthy, Nov 11 2002

			The 13th prime, 41, divides the repunit 11111, the smallest among all R(5k) which are multiples of 41.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
P. De Geest, Repunits and Their Factors
Amarnath Murthy, On the divisors of Smarandache Unary Sequence, Smarandache Notions Journal, Vol. 11, 2000.
A. A. A. Steward, Factorization of Repunits[up to R(196)]. Cached copy, Sep 25 2000.
Michael De Vlieger, Records and positions in first 1000 terms of A071126

Number of 1's in A077573(n).

Cf. A000042. Apart from a(2), identical to A002371.

Table[Function[p, If[Divisible[10, p], 0, k = {1}; While[! Divisible[ FromDigits@ k, p], AppendTo[k, 1]]; Length@ k]]@ Prime@ n, {n, 67}] (* Michael De Vlieger, May 20 2017 *)

from sympy import prime
from itertools import count
def a(n):
    if n == 1 or n == 3: return 0
    pn = prime(n)
    return next(k for k in count(1) if 10**k//9%pn == 0)
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Jul 24 2025

A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

Original entry on oeis.org

1, 1, 11, 11, 11, 11, 111, 111, 111, 111, 111, 111, 111, 111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111

Offset: 1

M. F. Hasler, Mar 21 2015

nonn

Yields the length of the n-th (nonempty) binary word (or word over any 2-letter alphabet, like A007931 or A032810 or A032834) in tally mark notation (A000042).

Felix Fröhlich, Table of n, a(n) for n = 1..10000

lim = 5; lst = Table[(10^n - 1)/9, {n, 0, lim}]; Reap@ For[i = 1, i <= lim, i++, Sow@ Table[lst[[i + 1]], {d, 2^i}]] // Flatten // Rest (* Michael De Vlieger, Apr 01 2015 *)

PARI
```
a(n)=10^#binary(n+1)\90
```

def A256077(n): return (10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Nov 07 2024

a(n) = A002275(A000523(n+1)) = A032810(n)-A007931(n) = A032834(n)-A032810(n), etc.

A063432 Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

Original entry on oeis.org

1, 11, 10, 111, 11, 10, 1111, 100, 11, 10, 11111, 101, 12, 11, 10, 111111, 110, 20, 12, 11, 10, 1111111, 111, 21, 13, 12, 11, 10, 11111111, 1000, 22, 20, 13, 12, 11, 10, 111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 1111111111, 1010, 101, 22, 20, 14, 13

Offset: 1

Henry Bottomley, Jul 20 2001

Representation of n in base 1 is defined to be a concatenation of n 1's.
It is difficult to write twenty-one in base 11 using decimal digits.
Representation in bases greater than 10 are written in base 10. This is really nasty! - N. J. A. Sloane, Dec 06 2002

			Rows start (1), (11, 10), (111, 11, 10), (1111, 100, 11, 10), etc.

Cf. A063431.
Rows are effectively the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717, etc.
Columns are truncated versions of A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649, etc.
Without the 1st column becomes A004053.

f[n_] := Flatten[ Append[ {FromDigits[ Table[1, {n}]] }, Table[ FromDigits[ IntegerDigits[n, i]], {i, 2, n}]]]; Flatten[ Table[ f[n], {n, 1, 10}]] (* Robert G. Wilson v *)

A081988 Product of digits + 1 is a prime.

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 21, 22, 23, 25, 26, 28, 29, 32, 34, 36, 41, 43, 44, 47, 49, 52, 56, 58, 61, 62, 63, 65, 66, 67, 74, 76, 82, 85, 89, 92, 94, 98, 111, 112, 114, 116, 121, 122, 123, 125, 126, 128, 129, 132, 134, 136, 141, 143, 144, 147, 149, 152, 156, 158

Offset: 1

Amarnath Murthy, Apr 04 2003

Sequence is infinite: every number of the form 111..1 is in the sequence, for example. - Gabriel Cunningham (gcasey(AT)mit.edu), Apr 07 2003

			126 is a member as 1*2*6 + 1 = 13 is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000042, A052382.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

Offset changed and a(60) and beyond from Michael S. Branicky, Aug 22 2022

cp's OEIS Frontend

A065444 Decimal expansion of 9*Sum_{k>=1} 1/(10^k-1).

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A086066 a(n) = Sum_{d in D(n)} 2^d, where D(n) = set of digits of n in decimal representation.

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A255805 Numbers with no zeros in base-8 representation.

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A248910 Numbers with no zeros in base-6 representation.

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A255808 Numbers with no zeros in base-9 representation.

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A322761 Irregular triangle read by rows in which n-th row lists all partitions of n, in graded reverse lexicographic ordering, using a compressed notation.

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Maple

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A071126 Length of least repunit which is a multiple of the n-th prime, or 0 if no such multiple exists.

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