A002282 -id:A002282 - cp's OEIS frontend

A352155 Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.

1, 6, 7, 8, 9, 10, 14, 24, 26, 28, 32, 35, 54, 55, 56, 60, 64, 65, 66, 70, 72, 74, 75, 80, 82, 88, 90, 100, 104, 112, 128, 140, 175, 176, 224, 240, 260, 280, 320, 350, 432, 448, 468, 504, 512, 528, 540, 548, 550, 560, 572, 576, 584, 592, 600, 616, 625, 640, 650, 660

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 17 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.
{8, 88, 888, ...} = A002282 \ {0} is a subsequence.

			m = 14 is a term since 1/14 = 0.0714285714285714285... and the smallest term after the leading 0 is 1.
m = 240 is a term since 1/240 = 0.00416666666... and the smallest term after the leading 0's is 1.
m = 888 is a term since 1/888 = 0.001126126126... and the smallest term after the leading 0's is 1.

Cf. A002282, A333402.

Similar with smallest digit k: A352154 (k=0), this sequence (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 1 &]

from itertools import count, islice
from sympy import multiplicity, n_order
def A352155_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '1':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '1':
                yield n
A352155_list = list(islice(A352155_gen(),20)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 1.

A180160 (sum of digits) mod (number of digits) of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2

Offset: 0

Reinhard Zumkeller, Aug 15 2010

a(n) = A007953(n) mod A055642(n);
a(A061383(n)) = 0; a(A180157(n)) > 0;
a(repdigits)=0: a(A010785(n))=0: a(A002275(n))=0: a(A002276(n))=0: a(A002277(n))=0: a(A002278(n))=0: a(4(n))=0: a(A002279(n))=0: a(A002280(n))=0: a(A002281(n))=0: a(A002282(n))=0: a(A002283(n))=0;
A123522 gives smallest m such that a(m) = n.

Reinhard Zumkeller, Table of n, a(n) for n = 0..25000

Cf. A002275-A002283, A007953, A010785, A055642, A061383, A123522, A180157, A374097.

A180160[n_] := If[n == 0, 0, Mod[Total[#], Length[#]] & [IntegerDigits[n]]];
Array[A180160, 100, 0] (* Paolo Xausa, Jun 30 2024 *)
Join[{0},Table[Mod[Total[IntegerDigits[n]],IntegerLength[n]],{n,110}]] (* Harvey P. Dale, Jul 30 2025 *)

A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

Original entry on oeis.org

1, 6, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least positive integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A249572 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

For n > 2, a(n) + a(n+1) divides the plane into 2 regions. For n > 1, a(2n) - a(2n-1) divides the plane into n+1 regions. For n >= 1, a(2n+1) - a(2n) divides the plane into n regions. - Ivan N. Ianakiev, Feb 23 2015

			The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller positive integer does this, so a(4) = 68.

Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A249572, A250257, A250258, A001743, A001744, A001745, A001746, A002282.

I:=[1,6,8,68]; [n le 4 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

Join[{1, 6, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 4.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 4.
G.f.: x*(10*x^3 - 8*x^2 + 5*x + 1)/((x - 1)*(10*x^2 - 1)). (End)
E.g.f.: (9 + 45*x - 40*cosh(x) + 31*cosh(sqrt(10)*x) - 40*sinh(x) + 4*sqrt(10)*sinh(sqrt(10)*x))/45. - Stefano Spezia, Aug 11 2025

A098406 a(n) = (10^n + 17)/9.

Original entry on oeis.org

2, 3, 13, 113, 1113, 11113, 111113, 1111113, 11111113, 111111113, 1111111113, 11111111113, 111111111113, 1111111111113, 11111111111113, 111111111111113, 1111111111111113, 11111111111111113, 111111111111111113, 1111111111111111113, 11111111111111111113, 111111111111111111113

Offset: 0

Klaus Brockhaus, Sep 07 2004

A097683 gives numbers k such that a(k) is prime.

			a(5) = (100000 + 17)/9 = 11113.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A097683, A047855, A002275.

Cf. A002282, A062397.

FromDigits/@Table[PadLeft[{3},n,1],{n,20}] (* Harvey P. Dale, Jun 18 2011 *)

PARI
```
for(n=1,18,print1(((10^n)+17)/9,","))
```

a(1) = 3; a(n) = a(n-1) + 10^(n-1).
a(1) = 3; a(n) = 10*a(n-1) - 17.
a(n) = A047855(n)+1 = A002275(n)+2.
G.f.: (2-19*x)/((10*x-1)*(x-1)). - R. J. Mathar, Jan 27 2017
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(x)*(exp(9*x) + 17)/9.
a(n) = A062397(n) - A002282(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1. (End)

a(0) from Ivan Panchenko, Nov 02 2013

A294327 a(n) = ((9n + 8)10^n - 8)/9.

Original entry on oeis.org

0, 18, 288, 3888, 48888, 588888, 6888888, 78888888, 888888888, 9888888888, 108888888888, 1188888888888, 12888888888888, 138888888888888, 1488888888888888, 15888888888888888, 168888888888888888, 1788888888888888888, 18888888888888888888, 198888888888888888888

Offset: 0

Seiichi Manyama, Oct 28 2017

Colin Barker, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A002282, A294328, A294329.

concat(0, Vec(18*x*(1 - 5*x) / ((1 - x)*(1 - 10*x)^2) + O(x^30))) \\ Colin Barker, Oct 28 2017

From Colin Barker, Oct 28 2017: (Start)
G.f.: 18*x*(1 - 5*x) / ((1 - x)*(1 - 10*x)^2).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n>2.
(End)

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.

Original entry on oeis.org

1, 818, 88188, 8881888, 888818888, 88888188888, 8888881888888, 888888818888888, 88888888188888888, 8888888881888888888, 888888888818888888888, 88888888888188888888888, 8888888888881888888888888, 888888888888818888888888888, 88888888888888188888888888888, 8888888888888881888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

A332181 := n -> 8*(10^(2*n+1)-1)/9-7*10^n;

Array[8 (10^(2 # + 1)-1)/9 - 7*10^# &, 15, 0]

apply( {A332181(n)=10^(n*2+1)\9*8-7*10^n}, [0..15])

def A332181(n): return 10**(n*2+1)//9*8-7*10**n

a(n) = 8*A138148(n) + 10^n = A002282(2n+1) - 7*10^n.
G.f.: (1 + 707*x - 1500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332187 a(n) = 8*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

7, 878, 88788, 8887888, 888878888, 88888788888, 8888887888888, 888888878888888, 88888888788888888, 8888888887888888888, 888888888878888888888, 88888888888788888888888, 8888888888887888888888888, 888888888888878888888888888, 88888888888888788888888888888, 8888888888888887888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077776-1)/2 = A183190: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332117 .. A332197 (variants with different "wing" digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

A332187 := n -> 8*(10^(2*n+1)-1)/9-10^n;

Array[8 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{7,878,88788},20] (* Harvey P. Dale, Jul 21 2024 *)

apply( {A332187(n)=10^(n*2+1)\9*8-10^n}, [0..15])

def A332187(n): return 10**(n*2+1)//9*8-10**n

a(n) = 8*A138148(n) + 7*10^n = A002282(2n+1) - 10^n.
G.f.: (7 + 101*x - 900*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A134778 Last digit of n alphabetically.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Nov 11 2007

Digits are decimal with names in English (see A000052). A134778(n)=A134777(n) iff n is a repdigit (n=A010785(m)), in which case a(n)=A010888(m), the repeated digit. a(n)=0 iff n is a member of A011540. a(n)=8 iff n is a member of A002282-{0}.

			a(104) = 0 because the digits of 104 are 1 (one), 0 (zero) and 4 (four) and "zero" occurs after both "four" and "one" alphabetically.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A134777, A000052, A010785, A010888, A011540, A002282.

def alpha(n): return [8, 5, 4, 9, 1, 7, 6, 3, 2, 0].index(n)
def a(n): return sorted(map(int, str(n)), key=alpha)[-1]
print([a(n) for n in range(105)]) # Michael S. Branicky, Dec 12 2023

A250257 Least nonnegative integer whose decimal digits divide the plane into n regions.

Original entry on oeis.org

1, 0, 8, 48, 88, 488, 888, 4888, 8888, 48888, 88888, 488888, 888888, 4888888, 8888888, 48888888, 88888888, 488888888, 888888888, 4888888888, 8888888888, 48888888888, 88888888888, 488888888888, 888888888888, 4888888888888, 8888888888888, 48888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Identical to A249572 except that a(2) = 0, not 4. See also the comments in A249572.

			The integer 48, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 48.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

Cf. A249572, A250256, A250258, A001743, A001744, A001745, A001746, A002282.

I:=[1,0,8,48,88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Nov 15 2014

Join[{1, 0, 8}, RecurrenceTable[{a[1]==48, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 5.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.
G.f.: x*(-40*x^4 + 50*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

A250258 Least nonnegative integer whose decimal digits divide the plane into n regions (A250257 variant).

Original entry on oeis.org

1, 0, 8, 68, 88, 688, 888, 6888, 8888, 68888, 88888, 688888, 888888, 6888888, 8888888, 68888888, 88888888, 688888888, 888888888, 6888888888, 8888888888, 68888888888, 88888888888, 688888888888, 888888888888, 6888888888888, 8888888888888, 68888888888888

Offset: 1

Rick L. Shepherd, Nov 15 2014

Equivalently, with offset 0, least nonnegative integer with n holes in its decimal digits. Leading zeros are not permitted. Variation of A250257 with the numeral "4" considered open at the top, as it is often handwritten. See also the comments in A249572.

			The integer 68, whose decimal digits have 3 holes, divides the plane into 4 regions. No smaller nonnegative integer does this, so a(4) = 68.

Index entries for linear recurrences with constant coefficients, signature (1, 10, -10).

Cf. A249572, A250256, A250257, A001743, A001744, A001745, A001746, A002282.

I:=[1,0,8,68,88]; [n le 5 select I[n] else 10*Self(n-2)+8: n in [1..40]]; // Vincenzo Librandi, Nov 16 2014

Join[{1, 0, 8}, RecurrenceTable[{a[1]==68, a[2]==88, a[n]==10 a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Nov 16 2014 *)

a(n) = 10*a(n-2) + 8 for n >= 5.
a(n) = A250256(n), n<>2.
From Chai Wah Wu, Jul 12 2016: (Start)
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n > 5.
G.f.: x*(-60*x^4 + 70*x^3 - 2*x^2 - x + 1)/((x - 1)*(10*x^2 - 1)). (End)

cp's OEIS Frontend

A352155 Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A180160 (sum of digits) mod (number of digits) of n in decimal representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A250256 Least positive integer whose decimal digits divide the plane into n regions (A249572 variant).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A098406 a(n) = (10^n + 17)/9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A294327 a(n) = ((9*n + 8)*10^n - 8)/9.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A332181 a(n) = 8*(10^(2n+1)-1)/9 - 7*10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A332187 a(n) = 8*(10^(2n+1)-1)/9 - 10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A134778 Last digit of n alphabetically.

A294327 a(n) = ((9n + 8)10^n - 8)/9.

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.