A047841 -id:A047841 - cp's OEIS frontend

A104785 a(n) = number of members of A047841 with n digits.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 35, 0, 21, 0, 0, 0, 20, 0, 15, 0, 6, 9, 1, 1

Offset: 1

Lekraj Beedassy, Mar 25 2005

Cf. A047841, A104786, A104787.

Edited by David Wasserman, Apr 17 2008

A104786 a(n) = least member of A047841 with n digits, or 0 if no such number exists.

Original entry on oeis.org

0, 22, 0, 0, 0, 0, 0, 10213223, 0, 1031223314, 0, 0, 0, 10413223241516, 0, 1051322314251617, 0, 106132231415261718, 1011112131415161718, 10713223141516271819, 101112213141516171819

Offset: 1

Lekraj Beedassy, Mar 25 2005

Cf. A047841, A104785, A104787, A039616, A039617.

Edited by David Wasserman, Apr 17 2008

A104787 a(n) = greatest member of A047841 with n digits, or 0 if no such number exists.

Original entry on oeis.org

0, 22, 0, 0, 0, 0, 0, 31331819, 0, 3122331819, 0, 0, 0, 41322324171819, 0, 5132232516171819, 0, 613223141526171819, 1111213141516171819, 10713223141516271819, 101112213141516171819

Offset: 1

Lekraj Beedassy, Mar 25 2005

Cf. A047841, A104785, A104786, A039616, A039617.

Edited by David Wasserman, Apr 17 2008

A104784 Erroneous version of A047841.

Original entry on oeis.org

22, 10213223, 10311233, 21322314, 21322315, 21322316, 21322317, 21322318, 21322319, 31123314, 31123315, 31123316, 31123317, 31123318, 31123319, 31331415, 31331416, 31331417, 31331418, 31331419, 31331516, 31331517, 31331518

Offset: 1

dead

J. N. Kapur, Reflections of a Mathematician, Chapter 33, pp. 314-318, Arya Book Depot, New Delhi 1996.

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

A046043 Autobiographical numbers (or curious numbers): list of numbers m = x_0 x_1 x_2 ... x_{b-1} (written in base b) such that x_i is the number of "digits" in m that are equal to i, for all i=0,...,b-1.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000

Offset: 1

Robert Leduc (leduc(AT)macalester.edu)

Note that the base b is the total number of "digits" in m. Since the numbers are written without spaces between the digits x_i, we must take b <= 10.
There are no such numbers for b<=3 or b=6, two such numbers for b=4, and exactly one such number for b=5 and each b>=7. - David Callan, Feb 17 2017
The proof of completeness is based on: x_0 > 0; x_i > 2 only if i = 0; for i > 2, x_i = 1 if i = x_0, x_i = 0 otherwise.
Enumerated by David Castro (david_castro(AT)retek.com).

			m = 1210 is written in base 4 (since it has 4 digits), and has one 0, two 1's, one 2 and zero 3's and m = "one two one zero".

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
M. Gardner, Mathematical Circus, pp. 128; 135 Prob. 7 Alfred A. Knopf NY 1979.
Tanya Khovanova, A Story of Storytelling Numbers, Math. Horizons, Sep 2009, 14-17.

Fred Gavin, Letters to the editor, Math. Mag 66 (4) (1993) p 276.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Amy Harmon, Beyond 'Hidden Figures': Nurturing New Black and Latino Math Whizzes, New York Times, Feb 17, 2017.
Tanya Khovanova, Autobiographical Numbers, arXiv:0803.0270 [math.CO], 2008.
Chris Smith, #MegaFavNumbers Self-Descriptive Numbers (the beauty 6210001000), video (2020)
K. Uhland, The Ten-Digit Number [Broken link?]

Cf. A138480 (another version), A359049.

Compare with the "Look-and-Say" version A047841.

isSelfDescribing[n_Integer] := (RotateRight[DigitCount[n]] == PadRight[IntegerDigits[n], 10]); Select[Range[10^10 - 1], isSelfDescribing] (* Martin Ettl, Oct 09 2012 *) (* Warning: This program causes Mathematica to crash! - David Callan, Feb 17 2017 *)

Edited by N. J. A. Sloane, Aug 15 2019

A267491 Autobiographical numbers in base 3: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10111, 11112, 100101, 1011122, 2012112, 2021102, 10010122, 10011112, 10101102

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 2 in that order when they occur in a number. The numbers and the frequency are written in base 3.
These are all autobiographical numbers in base 3 which lead to a fixed-point or belong to a cycle. However, in base 3 there is only one cycle with length 3, all other terms are fixed-points.
10011112, 10101102, 2012112 is a cycle of length 3, which means T(10011112)=10101102, T(10101102)=2012112, T(2012112)=10011112.

			11112 contains four 1's (11 in base 3), and one 2, so T(11112) = 11 1 1 2, and so 11112 is fixed under T.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267492 Autobiographical numbers in base 4: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 1011112, 1011113, 1111213, 10311233, 101112213

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 3 in that order when they occur in a number. The numbers and the frequency are written in base 4.

These are all autobiographical numbers in base 4 which lead to a fixed-point or belong to a cycle. However, in base 4 there are no cycles of length greater than 1, that is, all the terms are fixed-points.

			1011112 contains one 0, five (11 in base 4) 1's, and one 2, so T(1011112) = 1 0 11 1 1 2, and this is fixed under T.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht", 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491.

A267493 Autobiographical numbers in base 5: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10313314, 10311233, 21322314, 31123314, 101111314, 101111213, 101111214, 111121314, 1031223314, 10111221314

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 4 in that order when they occur in a number. The numbers and the frequency are written in base 5.

These are all autobiographical numbers in base 5 which lead to a fixed-point or belong to a cycle. However, in base 5 there are no cycles of length greater than 1, that is, all the terms are fixed-points.

			10213223 contains 1 0's, 2 1's, 3 2's and 2 3's.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A047842.

Cf. A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

A267494 Autobiographical numbers in base 6: numbers which are fixed or belong to a cycle under the operator T.

Original entry on oeis.org

22, 10213223, 10311233, 10313314, 10313315, 21322314, 21322315, 31123314, 31123315, 31331415, 1031223314, 1031223315, 3122331415, 10111121314, 10111121315, 10111121415, 10111131415, 11112131415, 103142132415, 104122232415, 1011122131415

Offset: 1

Antonia Münchenbach, Jan 16 2016

The T operator numerically summarizes the frequency of digits 0 through 5 in that order when they occur in a number. The numbers and the frequency are written in base 6.
These are all autobiographical numbers in base 6 which lead to a fixed-point or belong to a cycle.
There is one cycle with length 2 (103142132415, 104122232415), all other numbers are fixed-points.

			10213223 contains one 0, two 1's, three 2's and two 3's, so T(10213223) = 1 0 2 1 3 2 2 3, and this is fixed under T.
103142132415 and 104122232415 belong to the cycle of length 2, so
T(T(103142132415)) = T(1 0 4 1 2 2 2 3 2 4 1 5) = 1 0 3 1 4 2 1 3 2 4 1 5.

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Andre Kowacs, Studies on the Pea Pattern Sequence, arXiv:1708.06452 [math.HO], 2017.

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267502.

cp's OEIS Frontend

A104785 a(n) = number of members of A047841 with n digits.

Views

Author

Keywords

Crossrefs

Extensions

A104786 a(n) = least member of A047841 with n digits, or 0 if no such number exists.

Views

Author

Keywords

Crossrefs

Extensions

A104787 a(n) = greatest member of A047841 with n digits, or 0 if no such number exists.

Views

Author

Keywords

Crossrefs

Extensions

A104784 Erroneous version of A047841.

Views

Author

Keywords

References

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A046043 Autobiographical numbers (or curious numbers): list of numbers m = x_0 x_1 x_2 ... x_{b-1} (written in base b) such that x_i is the number of "digits" in m that are equal to i, for all i=0,...,b-1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A267491 Autobiographical numbers in base 3: numbers which are fixed or belong to a cycle under the operator T.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A267492 Autobiographical numbers in base 4: numbers which are fixed or belong to a cycle under the operator T.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A267493 Autobiographical numbers in base 5: numbers which are fixed or belong to a cycle under the operator T.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A267494 Autobiographical numbers in base 6: numbers which are fixed or belong to a cycle under the operator T.

Views