A010785 -id:A010785 - cp's OEIS frontend

A084034 Numbers which are a product of repeated-digit numbers A010785.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 111, 112, 120, 121, 125, 126, 128

Offset: 1

Amarnath Murthy, May 26 2003

Numbers which can be written as a*b*c*... where a, b, c are numbers whose decimal expansions are repetitions of a single digit.
Superset of A051038. The first numbers in this sequence but not in A051038 are 111, 222, 333, 444, 555. - R. J. Mathar, Sep 17 2008
From David A. Corneth, Aug 03 2017: (Start)
Closed under multiplication.
Multiples of 1-digit primes and numbers of the form (10^n - 1) / 9. (End)

			99 is a member since 99 = 3*33.
9768 is a member since 9768= 2*22*222.
111*2*33*44 = 322344 is a member.

David A. Corneth, Table of n, a(n) for n = 1..12917 (Terms <= 10^8)

Cf. A002275, A010785, A051038, A084348.

A002473 gives products of single-digit numbers.

isA010786 := proc(n) if nops(convert(convert(n,base,10),set)) = 1 then true; else false ; fi; end: isA084034 := proc(n,a010785) local d ; if n = 1 then RETURN(true) ; fi; for d in ( numtheory[divisors](n) minus{1} ) do if d in a010785 then if isA084034(n/d,a010785) then RETURN(true) ; fi; fi; od: RETURN(false) ; end: nmax := 1000: a010785 := [] : for k from 1 to nmax do if isA010786(k) then a010785 := [op(a010785),k] ; fi; od: for n from 1 to nmax do if isA084034(n,a010785) then printf("%d,",n) ; fi; end: # R. J. Mathar, Sep 17 2008

Corrected and extended by David Wasserman, Dec 09 2004
Corrected data, offset changed to 1 by David A. Corneth, Aug 03 2017
Edited by N. J. A. Sloane, Jul 02 2017 and Oct 10 2018

A139819 Complement of repdigit numbers A010785.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89

Offset: 1

N. J. A. Sloane, Jun 02 2008

Identical to (base 10) non-palindromic numbers A029742 up to a(83) = 101 which is a term of this sequence but not in A029742. - M. F. Hasler, Sep 08 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repdigit.
Wikipedia, Repdigit
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A066484 (subsequence).

Cf. A029742 (non-palindromic in base 10), A016038 (in any base), A050813 (in bases 2..10).

a139819 n = a139819_list !! (n-1)
a139819_list = filter ((== 0) . a202022) [0..] -- Reinhard Zumkeller, Dec 09 2011

isA139819 := proc(n)
    convert(n,base,10) ;
    convert(%,set) ;
    simplify(nops(%) >1 ) ;
end proc: # R. J. Mathar, Jan 17 2017

is_A139819(n)=#Set(digits(n))>1 \\ M. F. Hasler, Sep 08 2015

def A139819(n):
    m, k = n, n+9*((l:=len(str(n)))-1)+9*n//(10**l-1)
    while m != k:
        m, k = k, n+9*((l:=len(str(k)))-1)+9*k//(10**l-1)
    return m # Chai Wah Wu, Sep 04 2024

A202022(a(n)) = 0. - Reinhard Zumkeller, Dec 09 2011

A355698 a(n) is the number of repdigits divisors of n (A010785).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Jul 14 2022

More than the usual number of terms are displayed in order to show the difference from A087990.
The first 100 terms are the same first 100 terms of A087990, then a(101) = 1 while A087990(101) = 2, because 101 is the smallest palindrome that is not repdigit; the next difference is 121.
Inequalities: 1 <= a(n) <= A087990(n).

			66 has 8 divisors: {1, 2, 3, 6, 11, 22, 33, 66} that are all repdigits, hence a(66) = 8.
121 has 3 divisors: {1, 11, 121} of which 2 are repdigits: {1, 11}, hence a(121) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A010785, A065444, A087990, A190217, A340548, A355699.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

isrepdig:= proc(n) nops(convert(convert(n,base,10),set))=1 end proc:
f:= proc(n) nops(select(isrepdig, numtheory:-divisors(n))) end proc:
map(f, [$1..200]); # Robert Israel, Aug 07 2024

a[n_] := DivisorSum[n, 1 &, Length[Union[IntegerDigits[#]]] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 14 2022 *)

a(n) = my(ret=0,u=1); while(u<=n, ret+=sum(d=1,9, n%(u*d)==0); u=10*u+1); ret; \\ Kevin Ryde, Jul 14 2022

isrep(n) = {1==#Set(digits(n))}; \\ A010785
a(n) = sumdiv(n, d, isrep(d)); \\ Michel Marcus, Jul 15 2022

from sympy import divisors
def c(n): return len(set(str(n))) == 1
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 14 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (7129/2520) * A065444 = 3.11446261209177581335... . - Amiram Eldar, Apr 17 2025

A027828 First differences of A010785.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 11, 11, 11, 11, 11, 11, 11, 12, 111, 111, 111, 111, 111, 111, 111, 111, 112, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1112, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11112, 111111, 111111, 111111

Offset: 0

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

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

Cf. A010785.

Differences[Union[FromDigits/@Flatten[Table[PadRight[{},i,n],{n,0,9},{i,6}],1]]] (* Harvey P. Dale, Jan 16 2013 *)

More terms from James Sellers, May 01 2000

A070840 Repdigits (A010785) ordered by sum of digits (A007953).

Original entry on oeis.org

0, 1, 2, 11, 3, 111, 4, 22, 1111, 5, 11111, 6, 33, 222, 111111, 7, 1111111, 8, 44, 2222, 11111111, 9, 333, 111111111, 55, 22222, 1111111111, 11111111111, 66, 444, 3333, 222222, 111111111111, 1111111111111, 77, 2222222, 11111111111111, 555, 33333

Offset: 1

Amarnath Murthy, May 12 2002

If n has no divisors among 2,3,5 or 7 then the only term with digit sum n is the repunit A002275(n) = 1111... n times.

Cf. A070841.

Edited and extended by Ray Chandler, Feb 10 2009

A070841 Repdigits (A010785), excluding repunits (A002275), ordered by product of digits (A007954).

Original entry on oeis.org

2, 3, 4, 22, 5, 6, 7, 8, 222, 9, 33, 44, 2222, 55, 333, 22222, 66, 77, 88, 444, 222222, 99, 3333, 555, 2222222, 666, 33333, 4444, 22222222, 777, 888, 222222222, 5555, 999, 333333, 44444, 2222222222, 6666, 22222222222, 3333333, 7777, 55555, 8888

Offset: 1

Amarnath Murthy, May 12 2002

Cf. A070840.

Edited and extended by Ray Chandler, Feb 10 2009

A178049 The number of iterations of the map x -> x + d(x) to reach a repdigit (cf. A010785), starting at n, where the decimals of d(x) are the first differences of the decimals of x.

Original entry on oeis.org

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

Offset: 10

Michel Lagneau, May 18 2010

Let the recurrence X(k+1) = X(k) + Y(k) with the initial values : X(0) = n, and {x(1), x(2),...,x(p) } is the decimal expansion of n ; Y(0) has the decimal expansion {y(2), y(3),...,y(p)} where y(i) = abs(x(i)- x(i-1)), i = 2,..., p. For n > = 10, a(n) is the number of iterations of Y(k) needed to reach 0.

According to the computations with the Maple program for big numbers, the recurrence converges. Y(k) tends towards zero after a number of finite iterations, and X(k) tends towards a number q with the decimal expansion {p,p, ...,p }.

			a(11) = 1 because 11 + (1-1) = 11 + 0, and 0 is obtained after the first iteration.
a(12) = 7 because 12 + 1 = 13 -> 13 + 2 = 15 -> 15 + 4= 19 -> 19 + 8 = 27-> 27 + 5 = 32 -> 32 + 1 = 33 -> 33 + 0 = 33 is the last number of the cycle, and 0 is obtained after the 7th iteration.

Michel Lagneau, Table of n, a(n) for n = 10..10000

Cf. A010785.

for n from 10 to 200 do:n0:=n:s:=1:for i from 1 to 10^6 while(s<>0) do:x:=convert(n0,base,10):n1:=nops(x):s:=sum(abs(x[j+1]-x[j]),j=1..n1-1):n0:=n0+s:it:=i:od: printf(`%d, `, it):od:

A033618 Number of ways n-th repdigit number, A010785(n), can be expressed as a polygonal number.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 4, 5, 2, 3, 3, 4, 3, 4, 3, 4, 5, 3, 3, 3, 3, 2, 3, 3, 3, 6, 3, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 3, 7, 3, 4, 3, 4, 5, 4, 3, 4, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 3, 2, 3, 6, 2, 3, 3

Offset: 2

Mike Keith

			The n-th k-sided polygonal number is P(n,k)=n((k-2)n+4-k)/2 (k >= 2, n >= 1). For each repdigit number R>=2, sequence gives number of (n,k) such that P(n,k)=R.

M. Keith, On Repdigit Polygonal Numbers, J. Integer Sequences, Vol. 1, 1998, #6.

A277209 Partial sums of repdigit numbers (A010785).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 56, 78, 111, 155, 210, 276, 353, 441, 540, 651, 873, 1206, 1650, 2205, 2871, 3648, 4536, 5535, 6646, 8868, 12201, 16645, 22200, 28866, 36643, 45531, 55530, 66641, 88863, 122196, 166640, 222195, 288861, 366638, 455526, 555525, 666636, 888858, 1222191, 1666635, 2222190

Offset: 0

Ilya Gutkovskiy, Oct 05 2016

More generally, the ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1) is (Sum_{m = 1..(k-1)} m*x^m)/((1 - x)*(1 - x^(k-1))*(1 - k*x^(k-1))).

			a(0)=0;
a(1)=0+1=1;
a(2)=0+1+2=3;
a(3)=0+1+2+3=6;
...
a(10)=0+1+2+3+4+5+6+7+8+9+11=56;
a(11)=0+1+2+3+4+5+6+7+8+9+11+22=78;
a(12)=0+1+2+3+4+5+6+7+8+9+11+22+33=111, etc.

Eric Weisstein's World of Mathematics, Repdigit
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,11,-11,0,0,0,0,0,0,0,-10,10).

Cf. A000217, A010785, A027828, A046489.

CoefficientList[Series[x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 7 x^6 + 8 x^7 + 9 x^8)/((1 - x) (1 - 10 x^9) (1 - x^9)), {x, 0, 50}], x]

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8)/((1 - x)*(1 - x^9)*(1 - 10*x^9)).
a(n) = A000217(n) for n < 10.
a(n) = A046489(n-1) for n < 19.

A353704 Repdigit numbers (A010785) in A157037.

Original entry on oeis.org

6, 22, 66, 222, 555, 3333, 55555, 66666, 111111, 7777777, 2222222222, 5555555555, 55555555555555555, 2222222222222222222222222, 55555555555555555555555555, 66666666666666666666666666, 66666666666666666666666666666666666, 6666666666666666666666666666666666666666666

Offset: 1

Marius A. Burtea, May 08 2022

Intersection of A010785 and A157037.

No term contains the digits 4, 8 or 9.

			22 = A010785(11) and 22 = A157037(3), so 22 is a term.
66 = A010785(15) and 22 = A157037(8), so 66 is a term.

Cf. A002113, A003415, A010785, A157037.

f:=func;  [n:n in [(k - 9*Floor((k-1)/9))*(10^Floor((k+8)/9) - 1) div 9:k in [1..400]]| IsPrime(Floor(f(n))) ];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Sort[Flatten[Outer[Times, Range[1, 9], (10^Range[43] - 1)/9]]], PrimeQ[d[#]] &] (* Amiram Eldar, May 09 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = isprime(ad(m)) && (#Set(digits(m)) == 1); \\ Michel Marcus, May 09 2022

from itertools import count, islice
from sympy import isprime, factorint
def A353704_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), (d*(10**l-1)//9 for l in count(1) for d in (1,2,3,5,6,7)))
A353704_list = list(islice(A353704_gen(),10)) # Chai Wah Wu, Jun 23 2022

cp's OEIS Frontend

A084034 Numbers which are a product of repeated-digit numbers A010785.

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A139819 Complement of repdigit numbers A010785.

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A355698 a(n) is the number of repdigits divisors of n (A010785).

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A027828 First differences of A010785.

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A070840 Repdigits (A010785) ordered by sum of digits (A007953).

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A070841 Repdigits (A010785), excluding repunits (A002275), ordered by product of digits (A007954).

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A178049 The number of iterations of the map x -> x + d(x) to reach a repdigit (cf. A010785), starting at n, where the decimals of d(x) are the first differences of the decimals of x.

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A033618 Number of ways n-th repdigit number, A010785(n), can be expressed as a polygonal number.

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A277209 Partial sums of repdigit numbers (A010785).

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