A244112 -id:A244112 - cp's OEIS frontend

A278440 Numbers n such that A244112(n) | n.

22, 777, 4444, 68868, 200000, 303030, 333000, 333333, 555555, 660000, 660660, 666666, 700000, 2332200, 3131313, 4444400, 6060600, 7007000, 7700000, 9009790, 9656955, 9885585, 11517771, 14233221, 14331231, 14333110, 14411040, 15143331, 15233221, 15331231, 15333110

Offset: 1

Paolo P. Lava, Nov 25 2016

A244112(68868) = 3826 and 68868 / 3826 = 18.

Paolo P. Lava, Table of n, a(n) for n = 1..200

Cf. A237605, A244112, A278439, A278441.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do a:=sort(convert(n,base,10));
for k from 1 to trunc(nops(a)/2) do c:=a[k]; a[k]:=a[nops(a)-k+1]; a[nops(a)-k+1]:=c; od;  k:=1; b:=a[1]; c:=0;
for j from 2 to nops(a) do if a[j]=b then k:=k+1; else d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; k:=1; b:=a[j]; fi; od;
d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; if type(n/c,integer) then print(n); fi; od; end: P(10^99);

Select[Range[10^6], Divisible[#, FromDigits@ Flatten@ Map[IntegerDigits, DeleteCases[#, k_ /; First@ k == 0]] &@ Reverse@ MapIndexed[{#1, (First@ #2 - 1)} &, RotateRight@ DigitCount@ #]] &] (* Michael De Vlieger, Dec 12 2016 *)

A278441 Numbers n such that n | A244112(n).

Original entry on oeis.org

1, 2, 5, 10, 22, 26, 32, 62, 91, 330, 370, 519, 575, 710, 1060, 4055, 29377, 79554, 108690, 150320, 306440, 2510048, 3605570, 14233221, 14331231, 14333110, 14509410, 15143331, 15233221, 15331231, 15333110, 16143331, 16153331, 16233221, 16331231, 16333110, 17143331

Offset: 1

Paolo P. Lava, Nov 25 2016

The sequence is bounded. See comment in A278439.

			A244112(519) = 191511 and 191511 / 519 = 369.

Paolo P. Lava, List of terms and corresponding ratios for n <= 10^9.

Cf. A237605, A244112, A278439, A278440.

with(numtheory): P:=proc(q) local a,b,c,d,j,k,n; for n from 1 to q do a:=sort(convert(n,base,10));
for k from 1 to trunc(nops(a)/2) do c:=a[k]; a[k]:=a[nops(a)-k+1]; a[nops(a)-k+1]:=c; od;  k:=1; b:=a[1]; c:=0;
for j from 2 to nops(a) do if a[j]=b then k:=k+1; else d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; k:=1; b:=a[j]; fi; od;
d:=10*k+b; c:=c*10^(ilog10(d)+1)+d; if type(c/n,integer) then print(n); fi; od; end: P(10^99);

A343455 Numbers where A244112(n) = n.

Original entry on oeis.org

22, 14233221, 14331231, 14333110, 15143331, 15233221, 15331231, 15333110, 16143331, 16153331, 16233221, 16331231, 16333110, 17143331, 17153331, 17163331, 17233221, 17331231, 17333110, 18143331, 18153331, 18163331, 18173331, 18233221, 18331231, 18333110, 19143331

Offset: 1

J. Lowell, Apr 15 2021

Intersection of A278440 and A278441.

Fixed points of A244112.

			14233221 has one 4, two 3's, three 2's, and two 1's.

Martin Ehrenstein, Table of n, a(n) for n = 1..57

Cf. A244112, A278440, A278441.

a(9) and beyond from Martin Ehrenstein, Apr 16 2021

A010785 Repdigit numbers, or numbers whose digits are all equal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666

Offset: 0

N. J. A. Sloane

Complement of A139819. - David Wasserman, May 21 2008
Subsequence of A134336 and of A178403. - Reinhard Zumkeller, May 27 2010
Subsequence of A193460. - Reinhard Zumkeller, Jul 26 2011
Intersection of A009994 and A009996. - David F. Marrs, Sep 29 2018
Beiler (1964) called these numbers "monodigit numbers". The term "repdigit numbers" was used by Trigg (1974). - Amiram Eldar, Jan 21 2022

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, New York, 1964, p. 83.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric F. Bravo, Carlos A. Gómez and Florian Luca, Product of Consecutive Tribonacci Numbers With Only One Distinct Digit, J. Int. Seq., Vol. 22 (2019), Article 19.6.3.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Mahadi Ddamulira, Repdigits as sums of three balancing numbers, Mathematica Slovaca, (2019), hal-02405969.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, arXiv:2003.10705 [math.NT], 2020.
Mahadi Ddamulira, Tribonacci numbers that are concatenations of two repdigits, hal-02547159, Mathematics [math] / Number Theory [math.NT], 2020.
Mahadi Ddamulira, Padovan numbers that are concatenations of two distinct repdigits, Mathematica Slovaca, Vol. 71, No. 2 (2021), pp. 275-284.
Bart Goddard and Jeremy Rouse, Sum of two repdigits a square, arXiv:1607.06681 [math.NT], 2016. Mentions this sequence.
Bir Kafle, Florian Luca and Alain Togbé, Triangular Repblocks, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.
Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae, Vol. 52 (2020), pp. 137-145.
Benedict Vasco Normenyo, Bir Kafle, and Alain Togbé, Repdigits as Sums of Two Fibonacci Numbers and Two Lucas Numbers, Integers, Vol. 19 (2019), Article A55.
Salah Eddine Rihane and Alain Togbé, Repdigits as products of consecutive Padovan or Perrin numbers, Arab. J. Math., Vol. 10 (2021), pp. 469-480.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Eric Weisstein's World of Mathematics, Repdigit.
Wikipedia, Repdigit.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10).

Cf. A000005, A009994, A009996, A010879, A037904, A047842, A059995, A065444, A134336, A139819, A151949, A178401, A178403, A180410, A193459, A193460, A202022, A227362, A244112.

a010785 n = a010785_list !! n
a010785_list = 0 : r [1..9] where
   r (x:xs) = x : r (xs ++ [10*x + x `mod` 10])
-- Reinhard Zumkeller, Jul 26 2011

[(n-9*Floor((n-1)/9))*(10^Floor((n+8)/9)-1)/9: n in [0..50]]; // Vincenzo Librandi, Nov 10 2014

A010785 := proc(n)
    (n-9*floor(((n-1)/9)))*((10^(floor(((n+8)/9)))-1)/9) ;
end proc:
seq(A010785(n), n = 0 .. 100); # Robert Israel, Nov 09 2014

fQ[n_]:=Module[{id=IntegerDigits[n]}, Length[Union[id]]==1]; Select[Range[0,10000], fQ] (* Vladimir Joseph Stephan Orlovsky, Dec 29 2010 *)
Union[FromDigits/@Flatten[Table[PadRight[{},i,n],{n,0,9},{i,6}],1]] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,-10}, {0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88},40] (* Harvey P. Dale, Dec 28 2011 *)
Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 6}] (* Robert G. Wilson v, Oct 09 2014 *)
Table[(n - 9 Floor[(n-1)/9]) (10^Floor[(n+8)/9] - 1)/9, {n, 0, 50}] (* José de Jesús Camacho Medina, Nov 06 2014 *)

a(n)=10^((n+8)\9)\9*((n-1)%9+1) \\ Charles R Greathouse IV, Jun 15 2011

nxt(n,t=n%10)=if(t<9,n*(t+1),n*10+9)\t \\ Yields the term a(k+1) following a given term a(k)=n. M. F. Hasler, Jun 24 2016

is(n)={1==#Set(digits(n))}
inv(n) = 9*#Str(n) + n%10 - 9 \\ David A. Corneth, Jun 24 2016

def a(n): return 0 if n == 0 else int(str((n-1)%9+1)*((n-1)//9+1))
print([a(n) for n in range(55)]) # Michael S. Branicky, Dec 29 2021

print([0]+[int(d*r) for r in range(1, 7) for d in "123456789"]) # Michael S. Branicky, Dec 29 2021

# without string operations
def a(n): return 0 if n == 0 else (10**((n-1)//9+1)-1)//9*((n-1)%9+1)
print([a(n) for n in range(55)]) # Michael S. Branicky, Nov 03 2023

A037904(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2007
A178401(a(n)) > 0. - Reinhard Zumkeller, May 27 2010
From Reinhard Zumkeller, Jul 26 2011: (Start)
For n > 0: A193459(a(n)) = A000005(a(n)).
for n > 10: a(n) mod 10 = floor(a(n)/10) mod 10.
A010879(n) = A010879(A059995(n)). (End)
A202022(a(n)) = 1. - Reinhard Zumkeller, Dec 09 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=4, a(5)=5, a(6)=6, a(7)=7, a(8)=8, a(9)=9, a(10)=11, a(11)=22, a(12)=33, a(13)=44, a(14)=55, a(15)=66, a(16)=77, a(17)=88, a(n) = 11*a(n-9) - 10*a(n-18). - Harvey P. Dale, Dec 28 2011
A151949(a(n)) = 0; A180410(a(n)) = A227362(a(n)). - Reinhard Zumkeller, Jul 09 2013
a(n) = (n - 9*floor((n-1)/9))*(10^floor((n+8)/9) - 1)/9. - José de Jesús Camacho Medina, Nov 06 2014
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^9)*(1-10*x^9)). - Robert Israel, Nov 09 2014
A047842(a(n)) = A244112(a(n)). - Reinhard Zumkeller, Nov 11 2014
Sum_{n>=1} 1/a(n) = (7129/2520) * A065444 = 3.11446261209177581335... - Amiram Eldar, Jan 21 2022

Name clarified by Jon E. Schoenfield, Nov 10 2023

A047842 Describe n (count digits in order of increasing value, ignoring missing digits).

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1011, 21, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1012, 1112, 22, 1213, 1214, 1215, 1216, 1217, 1218, 1219, 1013, 1113, 1213, 23, 1314, 1315, 1316, 1317, 1318, 1319, 1014, 1114, 1214, 1314, 24, 1415, 1416

Offset: 0

N. J. A. Sloane

Digit count of n. The digit count numerically summarizes the frequency of digits 0 through 9 in that order when they occur in a number. - Lekraj Beedassy, Jan 11 2007

Numbers which are digital permutations of one another have the same digit count. Compare with first entries of "Look And Say" or LS sequence A045918. As in the latter, a(n) has first odd-numbered-digit entry occurring at n=1111111111 with digit count 101, but a(n) has first ambiguous term 1011. For digit count invariants, i.e., n such that a(n)=n, see A047841. - Lekraj Beedassy, Jan 11 2007

			a(31) = 1113 because (one 1, one 3) make up 31.
101 contains one 0 and two 1's, so a(101) = 1021.
a(131) = 2113.
For n = 20231231, the digits of the date 2023-12-31, last day of 2023, a(n) = 10213223 is a fixed point: a(a(n)) = a(n) (cf. A235775). Since a(n) is invariant under permutation of the digits of n (leading zeros avoided), this is independent of the chosen notation, yyyy-mm-dd or mm/dd/yyyy or dd.mm.yyyy. - _M. F. Hasler_, Jan 11 2024

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.
Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A005151, A047841, A047843, A127354, A127355.
Cf. A235775.
Cf. A244112 (the same but in order of decreasing value of digits), A010785.
Cf. A005150 (Look and Say: describe the number digit-wise instead of overall count).
Cf. A328447 (least m having the same digits as n).

import Data.List (sort, group); import Data.Function (on)
a047842 :: Integer -> Integer
a047842 n = read $ concat $
   zipWith ((++) `on` show) (map length xs) (map head xs)
   where xs = group $ sort $ map (read . return) $ show n
-- Reinhard Zumkeller, Jan 15 2014

dc[n_] :=FromDigits@Flatten@Select[Table[{DigitCount[n, 10, k], k}, {k, 0, 9}], #[[1]] > 0 &];Table[dc[n], {n, 0, 46}] (* Ray Chandler, Jan 09 2009 *)
Array[FromDigits@ Flatten@ Map[Reverse, Tally@ Sort@ IntegerDigits@ #] &, 46] (* Michael De Vlieger, Jul 15 2020 *)

A047842(n)={if(n, local(c=1, S="", d=vecsort(digits(n)), a(i)=Str(S, c, d[i])); for(i=2, #d, if(d[i]==d[i-1], c++, S=a(i-1); c=1)); eval(a(#d)), 10)} \\ M. F. Hasler, Feb 25 2018; edited Jan 10 2024

def A047842(n):
    s, x = '', str(n)
    for i in range(10):
        y = str(i)
        c = str(x.count(y))
        if c != '0':
            s += c+y
    return int(s) # Chai Wah Wu, Jan 03 2015

a(a(n)) = A235775(n). [By definition of A235775. - M. F. Hasler, Jan 11 2024]
a(A010785(n)) = A244112(A010785(n)). - Reinhard Zumkeller, Nov 11 2014
a(n) = a(A328447(n)) = a(m) for all n and all m having the same digits as n, with multiplicity. - M. F. Hasler, Jan 11 2024

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A036058 Summarize digits of preceding number, by decreasing digit value. Start with a(0) = 0.

Original entry on oeis.org

0, 10, 1110, 3110, 132110, 13123110, 23124110, 1413223110, 1423224110, 2413323110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110, 1433223110

Offset: 0

Floor van Lamoen

This kind of counting sequence is always eventually periodic with period 1, 2 or 3. - Herve Lehning (lehning(AT)noos.fr), Oct 01 2003

			The third term is 1110 because the second term contains one 1 and one 0.

Herve Lehning, Computer-aided or analytic proof?, College Mathematics Journal, Vol. 21, No. 3, 1990, pp. 228-239.
Index of eventually constant sequences.

Cf. A007890 (same as this, starting at 1), A001155 (same as this, but using method A047842: by increasing digit value), A005150 (as before, starting at 1), A036059 ("fibonacci" based on this), A036066.

a(n)=if(n>9,1433223110,[0,10,1110,3110,132110,13123110,23124110,1413223110, 1423224110,2413323110][n+1]) \\ Charles R Greathouse IV, Jul 24 2012

a(n,a=0)={for(k=1,n,a==(a=A244112(a))&&break);a} \\ M. F. Hasler, Feb 25 2018

a(n+1) = A244112(a(n)), a(0) = 0. - M. F. Hasler, Feb 25 2018

A007890 Summarize the previous term! (in decreasing order).

Original entry on oeis.org

1, 11, 21, 1211, 1231, 131221, 132231, 232221, 134211, 14131231, 14231241, 24132231, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221, 14233221

Offset: 1

Mira Bernstein

			For example, the term after 131221 is obtained by saying "one 3, two 2's, three 1's", which gives 13-22-31, i.e. 132231.

Onno M. Cain, Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.

Cf. A005150, A034003, A036058, A244112.

Nest[Append[#, FromDigits@ Flatten@ Map[Reverse, Tally@ ReverseSort@ IntegerDigits@ #[[-1]] ] ] &, {1}, 24] (* Michael De Vlieger, Jul 15 2020 *)

From Seiichi Manyama, Aug 18 2020: (Start)
a(1) = 1 and a(n) = A244112(a(n-1)) for n > 1.
a(n) = 14233221 for n >= 13. (End)

A036066 The summarize Lucas sequence: summarize the previous two terms, start with 1, 3.

Original entry on oeis.org

1, 3, 1311, 2331, 331241, 14432231, 34433241, 54533231, 2544632221, 163534435221, 263544436231, 363554634231, 463554733221, 17364544733221, 37263554634231, 37363554734231, 37364544933221, 1937263554933221, 3927263544835231, 391827264534836231, 293827363544836231

Offset: 0

Floor van Lamoen

After the 26th term the sequence goes into a cycle of 46 terms.

"Summarize" uses here method C = A244112: in order of decreasing digit value.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index to sequences related to say what you see

Cf. A036059.

Cf. A244112 (summarizing as used here: by decreasing digit value), A047842 (alternative summarizing method: by increasing digit value), A047843 (another method: don't omit missing digits between smallest and largest one).

a:= proc(n) option remember; `if`(n<2, 2*n+1, (p-> parse(cat(seq((c->
     `if`(c=0, [][], [c, 9-i][]))(coeff(p, x, 9-i)), i=0..9))))(
      add(x^i, i=map(x-> convert(x, base, 10)[], [a(n-1),a(n-2)]))))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 18 2022

a[0] = 1; a[1] = 3; a[n_] := a[n] = FromDigits @ Flatten @ Reverse @ Select[ Transpose @ { DigitCount[a[n-1]] + DigitCount[a[n-2]], Append[ Range[9], 0]}, #[[1]] > 0 &];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 30 2017 *)

{a=[1,3]; for(n=1,50,a=concat(a,A244112(eval(Str(a[n],a[n+1]))))); a} \\ M. F. Hasler, Feb 25 2018

a(n+1) = A244112(concat(a(n),a(n-1))). - M. F. Hasler, Feb 25 2018

A047843 Describe n: give frequency of each digit, by increasing size; mention also missing digits between the smallest and largest one.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1011, 21, 1112, 110213, 11020314, 1102030415, 110203040516, 11020304050617, 1102030405060718, 110203040506070819, 100112, 1112, 22, 1213, 120314, 12030415, 1203040516

Offset: 0

N. J. A. Sloane

Other methods to describe or summarize n are: A047842 (as here, but ignoring "missing" digits), A244112 (count digits in order of decreasing size, ignoring missing digits). - M. F. Hasler, Feb 25 2018

			131 contains two 1's, zero 2's and one 3, so a(131) = 210213.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A005151, A047841, A047842.

a[n_] := Module[{T, f}, T = Tally[IntegerDigits[n]]; f[_] = 0; Do[f[t[[1]]] = t[[2]], {t, T}]; Table[{f[k], k}, {k, Min@T[[All, 1]], Max@T[[All, 1]]} ] // Flatten // FromDigits];
a /@ Range[0, 26] (* Jean-François Alcover, Jan 07 2020 *)

A047843(n,S="")={if(n,for(d=vecmin(n=digits(n)),vecmax(n),S=Str(S,#select(t->t==d,n),d));eval(S),10)} \\ M. F. Hasler, Feb 25 2018

More accurate title from M. F. Hasler, Feb 25 2018

cp's OEIS Frontend

A278440 Numbers n such that A244112(n) | n.

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A278441 Numbers n such that n | A244112(n).

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A343455 Numbers where A244112(n) = n.

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A010785 Repdigit numbers, or numbers whose digits are all equal.

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A047842 Describe n (count digits in order of increasing value, ignoring missing digits).

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A036058 Summarize digits of preceding number, by decreasing digit value. Start with a(0) = 0.

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A007890 Summarize the previous term! (in decreasing order).

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