A010785 -id:A010785 - cp's OEIS frontend

A180599 Zero followed by infinitely many 9's.

0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Odimar Fabeny, Sep 10 2010

Another interpretation: A real number with an infinitesimally small difference from the integer 1 which is used to test the precision of calculating devices. - John W. Nicholson, Feb 01 2012

a(n) is also the number of n-digit positive repdigit numbers (A010785). - Stefano Spezia, Aug 15 2020

			Viewed as a real number: For a TI-89, entering 1.-10^-12 yields .999999999999; however, 1.-10^-13 yields 1. - _John W. Nicholson_, Feb 01 2012

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

Cf. A007953, A010785, A010888, A057427, A180592, A180594, A180595, A180596, A180597, A180598.

a[n_] := Mod[9 n - 1, 9] + 1; a[0] = 0; Array[a, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
PadRight[{0},120,9] (* Harvey P. Dale, May 03 2019 *)

a(n)=if(n,9,0) \\ Charles R Greathouse IV, Jan 04 2013

a(0) = 0, a(n) = 9 for n > 0.
a(n) = 9 * A057427(n).
a(n) = A010888(9*n), where A010888 is the digital root.
From Robert Israel, Dec 16 2014: (Start)
G.f.: 9*x/(1 - x).
E.g.f.: 9*(exp(x) - 1). (End)

More terms from Robert G. Wilson v, Sep 20 2010

Definition changed by N. J. A. Sloane, Feb 04 2012

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 133, 134, 135, 143, 145, 146, 147, 153, 154, 157, 158, 159, 164, 169, 174, 175, 185

Offset: 1

Reinhard Zumkeller, May 27 2010

A178401(a(n)) > 0; complement of A178402.
A010785, A050278, A178358, A178359 are subsequences;
a(n) = A131207(n) for n < 48;
a(n) = A134336(n) for n < 48;
a(n+1) = A032981(n) for n < 38.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A190217 Numbers all of whose divisors are repdigit numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 1111111111111111111, 2222222222222222222, 3333333333333333333, 4444444444444444444, 5555555555555555555, 6666666666666666666, 7777777777777777777, 8888888888888888888, 9999999999999999999

Offset: 1

Jaroslav Krizek, May 06 2011

Subset of A010785, A190220 and A190221.

			Number 99 is in sequence because all divisors of 99 (1, 3, 9, 11, 33, 99) are repdigit numbers.

A243535 Numbers whose list of divisors contains 2 distinct digits (in base 10).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 22, 31, 33, 41, 55, 61, 71, 77, 101, 113, 121, 131, 151, 181, 191, 199, 211, 311, 313, 331, 661, 811, 881, 911, 919, 991, 1111, 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 3313, 3331, 4111, 4441, 6661, 7177, 7717, 8111

Offset: 1

Jaroslav Krizek, Jun 13 2014

Numbers k such that A037278(k), A176558(k) and A243360(k) contain 2 distinct digits.
Many of the composite terms are in A203897. - Charles R Greathouse IV, Sep 06 2016
Terms are either repdigit numbers (A010785) or contain only 1 and a single other digit. - Michael S. Branicky, Nov 16 2022

			121 is in the sequence because the list of divisors of 121, i.e., (1, 11, 121), contains 2 distinct digits (1, 2).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4317 terms from Robert Israel)

Cf. A095048, A037278, A176558, A243360.
Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050.
Cf. A243543 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1..10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=2;A(n)); Arrangement of column B]

dmax:= 6: # get all terms of <= dmax digits
Res:= {}:
for a in [0,$2..9] do
    S:= {0}:
    for d from 1 to dmax do
        S:= map(t -> (10*t+1,10*t+a), S);
        Res:= Res union select(filter, S)
    od
od:
sort(convert(Res,list)): # Robert Israel, Sep 05 2016

Select[Range[9000],Length[Union[Flatten[IntegerDigits/@Divisors[ #]]]] == 2&] (* Harvey P. Dale, Dec 14 2017 *)

isok(n) = vd = []; fordiv(n, d, vd = concat(vd, digits(d))); #Set(vd) == 2; \\ Michel Marcus, Jun 13 2014

from sympy import divisors
from itertools import count, islice, product
def ok(n):
    s = set("1"+str(n))
    if len(s) > 2: return False
    for d in divisors(n, generator=True):
        s |= set(str(d))
        if len(s) > 2: return False
    return len(s) == 2
def agen():
    yield from [2, 3, 5, 7]
    for d in count(2):
        s = set()
        for first, other in product("123456789", "0123456789"):
            for p in product(sorted(set(first+other)), repeat=d-1):
                if other not in p: continue
                t = int(first+"".join(p))
                if ok(t): s.add(t)
        yield from sorted(s)
print(list(islice(agen(), 52))) # Michael S. Branicky, Nov 16 2022

A048328 Numbers that are repdigits in base 3.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 40, 80, 121, 242, 364, 728, 1093, 2186, 3280, 6560, 9841, 19682, 29524, 59048, 88573, 177146, 265720, 531440, 797161, 1594322, 2391484, 4782968, 7174453, 14348906, 21523360, 43046720, 64570081, 129140162, 193710244, 387420488, 581130733

Offset: 0

Patrick De Geest, Feb 15 1999

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Repdigit.
Index entries for 3-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Bisections: A024023 and A003462.

Cf. A000225, A010785, A028987, A028988, A033016, A038754, A068911, A124302, A214369.

nmax := 35; a(0) := 0: for n from 1 to nmax do a(2*n) := a(2*n-2) + 2*3^(n-1); od: a(1) := 1: for n from 1 to nmax do a(2*n+1) := 1*a(2*n-1) + 3^n; od: seq(a(n), n=0..nmax);
# End program 1
with(GraphTheory): G := PathGraph(5): A:= AdjacencyMatrix(G): nmax := nmax; for n from 1 to nmax+1 do B(n) := A^n; b(n) := add(B(n)[1, k], k=1..4); a1(n-1) := b(n)-1; od: seq(a1(n), n=0..nmax);
# End program 2
# From Johannes W. Meijer, May 29 2010, revised Sep 23 2012
# third Maple program:
a:= n->(<<0|1>, <-3|4>>^iquo(n, 2, 'r').`if`(r=0, <<0, 2>>, <<1, 4>>))[1, 1]:
seq (a(n), n=0..60);  # Alois P. Heinz, Sep 23 2012

Rest[FromDigits[#, 3]&/@Flatten[Table[{PadRight[{1}, n, 1], PadRight[{2}, n, 2]}, {n, 0, 20}], 1]] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -3,0,4,0]^n*[0;1;2;4])[1,1] \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (2*x^2+x)/(1-4*x^2+3*x^4). - Alois P. Heinz, Sep 23 2012
Sum_{n>=1} 1/a(n) = 3 * A214369 = 2.04646050781571420028... - Amiram Eldar, Jan 21 2022
a(n) = (3^(n/2)*(sqrt(3) + 2 - (-1)^n*(sqrt(3) - 2)) - 3 - (-1)^n)/4. - Stefano Spezia, Feb 18 2022

A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

Original entry on oeis.org

3, 5, 109, 111111109, 259259257

Offset: 1

Patrick De Geest, Apr 15 2000

No additional terms below 472882027.

No additional terms below 10^58. - Chai Wah Wu, Jun 01 2024

			a(5) is ok since between 259259257 and nextprime 259259261 we get the sum 259259258 + 259259259 + 259259260 which yield repdigit 777777777.

Eric Weisstein's World of Mathematics, Repdigit

Cf. A010785, A028987, A028988, A046933, A054264, A054265, A054266, A054267.

repQ[n_]:=Count[DigitCount[n],0]==9; Select[Prime[Range[2,14500000]], repQ[Total[Range[#+1,NextPrime[#]-1]]]&] (* Harvey P. Dale, Jan 29 2011 *)

from sympy import prime
A054268 = [prime(n) for n in range(2,10**5) if len(set(str(int((prime(n+1)-prime(n)-1)*(prime(n+1)+prime(n))/2)))) == 1]
# Chai Wah Wu, Aug 12 2014

from itertools import count, islice
from sympy import isprime, nextprime
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A054268_gen(): # generator of terms
    for l in count(1):
        c = []
        for m in range(1,10):
            k = m*(10**l-1)//9<<1
            for a, b in diop_quadratic((x-y-1)*(x+y)-k):
                if isprime(b) and a == nextprime(b):
                    c.append(b)
        yield from sorted(c)
A054268_list = list(islice(A054268_gen(),5)) # Chai Wah Wu, Jun 01 2024

Numbers A000040(n) for n > 1 such that A001043(n)*(A001223(n)-1)/2 is in A010785. - Chai Wah Wu, Aug 12 2014

Offset changed by Andrew Howroyd, Aug 14 2024

A244112 Reverse digit count of n in decimal representation.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1110, 21, 1211, 1311, 1411, 1511, 1611, 1711, 1811, 1911, 1210, 1211, 22, 1312, 1412, 1512, 1612, 1712, 1812, 1912, 1310, 1311, 1312, 23, 1413, 1513, 1613, 1713, 1813, 1913, 1410, 1411, 1412, 1413, 24, 1514, 1614, 1714

Offset: 0

Reinhard Zumkeller, Nov 11 2014

Frequencies of digits 0 through 9, occurring in n, are summarized in order of decreasing digits;

a(A010785(n)) = A047842(A010785(n)).

			101 contains two 1s and one 0, therefore a(101) = 2110;
102 contains one 2, one 1 and one 0, therefore a(102) = 121110.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A047842, A010785.

See A036058 for the orbit of 0 under this map.

import Data.List (sort, group); import Data.Function (on)
a244112 :: Integer -> Integer
a244112 n = read $ concat $
   zipWith ((++) `on` show) (map length xs) (map head xs)
   where xs = group $ reverse $ sort $ map (read . return) $ show n

f[n_] := Block[{s = Split@ IntegerDigits@ n}, FromDigits@ Reverse@ Riffle[Union@ Flatten@ s, Length@# & /@ s]]; Array[f, 48, 0] (* Robert G. Wilson v, Dec 01 2016 *)

A244112(n,c=1,S="")={for(i=2,#n=vecsort(digits(n),,4),n[i]==n[i-1]&&c++&&next;S=Str(S,c,n[i-1]);c=1);eval(Str(S,c,if(n,n[#n])))} \\ M. F. Hasler, Feb 25 2018

def A244112(n):
    return int(''.join([str(str(n).count(d))+d for d in '9876543210' if str(n).count(d) > 0])) # Chai Wah Wu, Dec 01 2016

A329197 Length of the n-th nontrivial cycle of the "ghost iteration" A329200.

Original entry on oeis.org

5, 6, 3, 7, 5, 9, 6, 3, 3, 5, 3, 3, 6, 3, 3, 3, 5, 3, 3, 6, 3, 3, 3, 3, 5, 3, 3, 6, 3, 17, 3, 11, 3, 3, 3, 5, 3, 3, 6, 6, 3, 17, 3, 3, 3, 3, 5, 7, 6

Offset: 1

M. F. Hasler, Nov 10 2019

A329200 consists in adding or subtracting the number A040115(n) whose digits are the differences of adjacent digits of n, depending on its parity.

Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence gives the length of these cycles, ordered by their smallest member, as they are listed in the table A329196. See there for more information.

			The first cycle of A329200 is row 1 of A329196, (8290, 8969, 9102), of length 3 = a(1).
The second cycle of A329200 is row 2 of A329196, (17998, 24199, 21819, 20041, 22084, 21800, 20020), of length 7 = a(2).

Cf. A329196, A329200, A329198, A329342 (variant using A329201).

/* change T to #T in print statement of code for A329196 */

a(9)-a(35) from Scott R. Shannon, Nov 12 2019

a(36)-a(49) from Lars Blomberg, Nov 15 2019

A329201 The ghost iteration (B): add or subtract the number formed by absolute differences of digits (A040115), according to parity (odd or even).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 13, 11, 17, 11, 21, 11, 25, 11, 18, 22, 22, 24, 22, 28, 22, 32, 22, 36, 33, 29, 33, 33, 35, 33, 39, 33, 43, 33, 36, 44, 40, 44, 44, 46, 44, 50, 44, 54, 55, 47, 55, 51, 55, 55, 57, 55, 61, 55, 54, 66, 58, 66, 62, 66, 66, 68, 66, 72, 77, 65, 77, 69, 77, 73, 77

Offset: 0

Eric Angelini and M. F. Hasler, Nov 09 2019

Sequence A040115 is most naturally extended to 0 (empty sum) for single-digit arguments; that's what we use for n < 10 here. This value is subtracted from n if even, added if odd.
A040115 is zero iff the argument is a repdigit (A010785), which therefore are the fixed points of this map A329201. All small starting values reach a fixed point, but larger values may enter a nontrivial cycle (or "loop").
See the table A329342 for the list of these cycles.

			For n = 101, the number formed by the absolute differences of digits is 11. Since this is odd it is added to n, so a(101) = 101 + 11 = 112.

E. Angelini, The ghost iteration, SeqFan list, Nov 2019.
E. Angelini, The ghost iteration, SeqFan list, Nov 2019 [Cached copy, with permission]
E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019.
E. Angelini, The Ghost Iteration, Personal blog "Cinquante signes", Nov 2019 [Cached copy, pdf file, with permission]

Cf. A040115, A329200 (variant A: add/subtract if even/odd), A010785 (fixed points).

Cf. A329342 (list of cycles).

apply( A329201(n)={n-(-1)^(n=fromdigits(abs((n=digits(n+!n))[^-1]-n[^1])))*n}, [1..199])

a(n) = n - (-1)^d*d where d = A040115(n), 0 for n < 10.

A340548 Integers whose number of repdigit divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 3960, 14652, 26664, 29304, 79992, 146520, 399960, 1025640, 2666664, 7999992, 13333320, 39999960, 269333064, 807999192, 1346665320, 4039995960, 28279971720, 7999999999992, 8080799919192, 13333333333320, 13467999865320, 39999999999960, 40403999595960

Offset: 1

Bernard Schott, Jan 11 2021

The first 10 terms are the same as A093036, then A093036(11) = 1848 while a(11) = 3960, because from a(1) to a(10), all palindromic divisors are also repdigits, and then 616 is a non-repdigit palindromic divisor of 1848.
Number of repdigit divisors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 17, 18, ...
Indices of repdigits: 1, 2, 3, 4, 7, ...

			132 has 12 divisors: {1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132} of which 10 are repdigits: {1, 2, 3, 4, 6, 11, 22, 33, 44, 66}. No positive integer smaller than 132 has as many as ten repdigit divisors; hence 132 is a term.

David A. Corneth, Table of n, a(n) for n = 1..53 (terms <= 10^30).

Cf. A010785, A190217.

Similar for: A053624 (odd), A181808 (even), A093036 (palindromes), A340549 (repunits).

repQ[n_] := Length @ Union @ IntegerDigits[n] == 1; s[n_] := DivisorSum[n, 1 &, repQ[#] &]; smax =  0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 11 2021 *)

isrd(n) = {1 == #Set(digits(n))}; \\ A010785
f(n) = sumdiv(n, d, isrd(d));
lista(nn) = {my(m = 0); for (n=1, nn, my(x = f(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 11 2021

a(16)-a(20) from Michel Marcus, Jan 11 2021
a(21)-a(26) from Amiram Eldar, Jan 12 2021
a(27) from Chai Wah Wu, Jan 14 2021
More terms from David A. Corneth, Jan 15 2021

cp's OEIS Frontend

A180599 Zero followed by infinitely many 9's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

Views

Author

Keywords

Comments

Links

A190217 Numbers all of whose divisors are repdigit numbers.

Views

Author

Keywords

Comments

Examples

A243535 Numbers whose list of divisors contains 2 distinct digits (in base 10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Excel

Maple

Mathematica

PARI

Python

A048328 Numbers that are repdigits in base 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

Extensions

A244112 Reverse digit count of n in decimal representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

A329197 Length of the n-th nontrivial cycle of the "ghost iteration" A329200.