A046043 -id:A046043 - cp's OEIS frontend

A138481 Mutually-praising pairs excluding autobiographical numbers A046043. Version 1: both numbers in a pair have at most 10 digits.

130, 230, 430, 530, 630, 730, 1101, 2210, 10110, 11200, 23100, 43100, 53100, 63100, 211100, 230100, 311100, 411100, 430100, 511100, 530100, 611100, 630100, 711100, 1001100, 2300100, 4211000, 4300100, 5211000, 5300100, 6211000, 6300100

Offset: 1

Tanya Khovanova, Mar 18 2008

			130 and 1101 are a mutually-praising pair: 130 specifies a number with 1 digit equal to "0", 3 digit "1" and nothing else. 1101 specifies 1 digit "0", 1 digit "1", no digits "2" and 1 digit "3".

Martin Fuller, Table of n, a(n) for n = 1..74
Tanya Khovanova, Autobiographical Numbers

Cf. A046043, A138482.

More terms from Joshua Zucker, Mar 19 2008 and Martin Fuller, Mar 20 2008

A138480 Autobiographical numbers: the first digit specifies how many 0's in the number, the next digit specifies how many 1's, etc. This version is not limited to 10 digits.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 72100001000, 821000001000, 9210000001000

Offset: 1

Martin Fuller, Mar 20 2008

			72100001000 has 7 digits equal to "0", 2 digits equal to "1", 1 digit each "2" and "7" and no other digits.

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Tanya Khovanova, Can you solve the Leonardo da Vinci riddle?, TED-Ed YouTube channel.

Autobiographical numbers up to 10 digits: A046043. Mutually-praising pairs: A138481 and A138482.

A108551 Self-descriptive numbers in various bases represented in base 10.

Original entry on oeis.org

100, 136, 1425, 389305, 8946176, 225331713, 6210001000, 186492227801, 6073061476032, 213404945384449, 8054585122464440, 325144322753909625, 13983676842985394176, 638488718313248327681, 30852387539151417415368, 1573159469597805848539033, 84423475200000000000008000

Offset: 1

Alonso del Arte, Jun 07 2005

A self-descriptive number in base b has b digits, indexed by 0 ... b-1 and for all n, the n-th digit equals the number of n's in the number. In base 10 there is exactly one such number, 6210001000.
From Iain Fox, Sep 16 2020: (Start)
(b-4)*b^(b-1) + 2*b^(b-2) + b^(b-3) + b^3 is in this sequence for b=4 and b>6.
For b>6, there exists exactly one self-descriptive number of base b. This number is of the form stated in the comment above.
Proof: A number in this sequence is of the form x_0*b^(b-1) + x_1*b^(b-2) + ... + x_{b-2}*b + x_{b-1} where x_i is an integer on the interval [0, b-1] and represents the number of times i appears in the sequence d = x_0, x_1, ..., x_{b-1}. Trivially, x_0 > 0. Let p be the number of nonzero terms in the sequence d. It is easy to see that p = Sum_{i=1..b-1} x_i. Since x_0 > 0, the number of nonzero terms in the sequence e = x_1, x_2, ..., x_{b-1} is p-1. Since the sum of the terms of e is one more than the number of nonzero terms in e, one term of e is 2 and the rest are either 0 or 1. This means that the only terms that can be in d are 0, 1, 2, and x_0, and thus there can be a maximum of four nonzero terms in d. Since there is a maximum of four nonzero terms in d, it is trivial that, for b>6, x_0>2. Thus, for b>6, there are exactly four nonzero terms in d. It is simple to determine that the four nonzero terms are x_0 = b-4, x_1 = 2, x_2 = 1, and x_{x_0} = x_{b-4} = 1.
(End)

			1210_4 = 100, 2020_4 = 136, 21200_5 = 1425, 3211000_7 = 389305,
42101000_8 = 8946176, 521001000_9 = 225331713, 6210001000_10,
72100001000_11 = 186492227801, 821000001000_12 = 6073061476032,
9210000001000_13 = 213404945384449, (10)2100000001000_14 =
8054585122464440, (11)21000000001000_15 = 325144322753909625,
(12)21000000001000_16 = 13983676842985394176, etc.

Clifford Pickover, Keys to Infinity, Chapter 28, "Chaos in Ontario." New York: Wiley, pp. 217-219, 1995.

Iain Fox, Table of n, a(n) for n = 1..350
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Eric Weisstein's World of Mathematics, Self-Descriptive Number.
Wikipedia, Self-descriptive number

Cf. A046043. See A274943 for another version.

Do[id = IntegerDigits[n, base]; If[id == (Count[id, # ] & /@ Range[0, base - 1]), Print[n]], {base, 2, 10}, {n, base^(base - 1), (base^base) - 1, base}]
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) fQ[lst_] := (lst == (Count[lst, # ] & /@ Range[0, Length[lst] - 1])); f[n_] := Block[{pts = PadLeft[ #, n] & /@ Partitions[n], k = 1, l = PartitionsP[n], lst = {}}, While[k < l, AppendTo[ lst, FromDigits[ Flatten[ Select[ Permutations[ pts[[k]]], fQ[ # ] &]], n]]; k++ ]; Drop[ Union[ lst], 1]]; Table[ f[n], {n, 2, 15}] (* Robert G. Wilson v, Jun 07 2005 *)

a(n)=if(n<4, [100, 136, 1425][n], (n+2)*(n*(n+3)-1)*(n+3)^n+(n+3)^3) \\ Iain Fox, Sep 16 2020

From Iain Fox, Sep 16 2020: (Start)
For n>3,
a(n) = (n-1)*(n+3)^(n+2) + 2*(n+3)^(n+1) + (n+3)^n + (n+3)^3 = (n+2)*(n*(n+3)-1)*(n+3)^n + (n+3)^3.
a(n) = A274943(n+3).
(End)

a(14)-a(17) from Iain Fox, Sep 16 2020

A138482 Mutually-praising pairs excluding autobiographical numbers A138480. Version 2: numbers may have more than 10 digits.

Original entry on oeis.org

130, 230, 430, 530, 630, 730, 830, 930, 1101, 2210, 10110, 11200, 23100, 43100, 53100, 63100, 73100, 83100, 93100, 211100, 230100, 311100, 411100, 430100, 511100, 530100, 611100, 630100, 711100, 730100, 811100, 830100, 911100, 930100

Offset: 1

Martin Fuller, Mar 20 2008

			130 and 1101 are a mutually-praising pair: 130 specifies a number with 1 digit equal to "0", 3 digit "1" and nothing else. 1101 specifies 1 digit "0", 1 digit "1", no digits "2" and 1 digit "3". 73100 and 21010001000 is the first pair with a number of more than 10 digits.

Martin Fuller, Table of n, a(n) for n = 1..172
Tanya Khovanova, Autobiographical Numbers

Cf. A046043, A138480, A138481.

A109167 a(n) = least nonnegative integer such that n appears a(n) times.

Original entry on oeis.org

1, 2, 1, 0, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18

Offset: 0

Rick L. Shepherd, Jun 20 2005

The concatenation of the first four terms' (decimal) digits gives 1210 = A046043(1).

			Note that we do not need to specify a(0) explicitly. Indeed, a(0) = 1 as it cannot be 0. a(1) = 2 as it cannot be 0 or 1.

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

Cf. A046043 (Autobiographical numbers), A001462 (Golomb's sequence).

from itertools import count, islice
def agen(): # generator of terms
    alst = [1, 2, 1, 0, 4, 4, 4, 4]
    yield from alst
    for n in count(5):
        add = [n]*alst[n]
        yield from add
        alst.extend(add)
print(list(islice(agen(), 79))) # Michael S. Branicky, Aug 25 2025

A359049 Autobiographical numbers k whose decimal digits are a concatenation count(0), count(1), ..., count(m) for some m, where count(j) is the number of (possibly overlapping) occurrences of j within the digits of k itself.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 1040110000031000, 9321000001201000, 94201000012110000, 1160010100041000010, 11611001000320000100, 13313000000001200000, 13313000000100200000

Offset: 1

Marc Morgenegg, Dec 14 2022

In other words: Counting the zeros (j=0) in the term gives the first concatenation of decimal digits (number of zeros) in the term, counting all ones, gives the second, and so on.
A term can have any number of digits.
This sequence is in base 10.

			1040110000031000 is a term: we have ten 0's, four 1's, zero 2's, one 3, one 4, three 10's and one 11 as integers in the term, visualized as follows:
  Integers(j): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
  term:       10 4 0 1 1 0 0 0 0 0  3  1  0  0  0
Notice that overlapping integers are counted so 110 is one 11, one 10 (or 111 would be two 11's).

Michael S. Branicky, Table of n, a(n) for n = 1..30
Michael S. Branicky, Python Program

Cf. A046043, A138480.

a(17)-a(20) from Michael S. Branicky, Dec 14 2022

A226195 Numbers n = x0 x1 x2...x9 such that xi is the number of digits greater than i in n.

Original entry on oeis.org

1, 10, 21, 22, 100, 210, 220, 311, 321, 332, 333, 1000, 2100, 2200, 3110, 3210, 3320, 3330, 4111, 4211, 4321, 4331, 4422, 4432, 4443, 4444, 10000, 21000, 22000, 31100, 32100, 33200, 33300, 41110, 42110, 43210, 43310, 44220, 44320, 44430, 44440, 51111, 52111

Offset: 1

Michel Lagneau, May 30 2013

This sequence contains 1142 terms.
Extension of the autobiographical numbers (or curious numbers) (A046043).
The concatenated numbers from n down to 1 (A000422(1) - A000422(9)): 1, 21, 321, ..., 987654321 are in the sequence.
The sequence includes A000461(n) (the n-digit number consisting of all n's) for n=1..9, i.e., 1, 22, 333, 4444, ..., 999999999.
The powers of 10 (A011557(0)..A011557(9)) are in the sequence.
The numbers of the form a(n)*10^p are also in the sequence.

			x0 x1 x2 x3 = 4211 is in the sequence because, for i = 0, 1, 2, 3:
xi > 0 (4 times) => x0 = 4;
xi > 1 (2 times) => x1 = 2;
xi > 2 (1 time) => x2 = 1;
xi > 3 (1 time) => x3 = 1.

Michel Lagneau, Table of n, a(n) for n = 1..1142

Cf. A000422, A000461, A011557, A046043.

T:=array(1..10):for n from 1 to 100000 do:nn:=length(n):for a from 1 to nn do:T[a] :=0:od:x:=convert(n,base,10): for k from 1 to nn do:for i from 1 to nn do: if k-1

A234512 Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.

Original entry on oeis.org

110, 311000, 2301000, 3003000, 3120000, 42100000, 410300000, 430100000

Offset: 1

Michel Lagneau, Dec 27 2013

In the decimal system a differential autobiographical number is a natural number such that d(0) is the number of differences |d(i)-d(i-1)| = 0, d(1) is the number of differences |d(i)-d(i-1)| = 1, and so on.
Property of this sequence: the sum of the decimal digits of a(n) equals length(a(n))-1.
It is possible to extend this problem by counting the differences |d(i)-d(i-1)| with the additional difference |d(r)-d(1)|. So we find a new sequence b(n) = 22100, 311100, 3022000, 20402000, 31310000, 40004000, 422010000, 430110000 with the property that the sum of the decimal digits of b(n) equals length(b(n)).

			311000 is in the sequence because the differential digits are:
|1-3| = 2;
|1-1| = 0;
|0-1| = 1;
|0-0| = 0;
|0-0| = 0, and
0 appears three times => 3;
1 appears one time => 1;
2 appears one time  => 1;
3 appears zero time => 0;
4 appears zero time => 0;
5 appears zero time => 0, hence a(2) = 311000.

Tanya Khovanova, Autobiographical Numbers

Cf. A037904, A046043, A108551, A138480.

with(numtheory):for n from 10 to 10^10 do:T:=array(0..9):for k from 0 to 9 do:T[k]:=0:od:x:=convert(n,base,10):n1:=nops(x):for i from 1 to n1-1 do:a:=abs(x[i]-x[i+1]):T[a]:=T[a]+1:od:s:=sum('T[i]*10^(10-i-1)','i'=0..9): for u from 9 by -1 to 1 do:if T[0]<>0 and irem(s,10^u)=0 and s/10^u = n then print(n):else fi:od:od:

A358538 Autobiographical numbers: the first digit of the term counts how many 0's the term in total has, the second how many 1's etc. up to the last digit but no more than b-1, where b is the base of the sequence. This sequence is in base b=10.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 52200100019, 52200100108, 52200101007, 52200110006, 52201100004, 52210100003, 53010100019, 53010100108, 53010101007, 53010110006, 53011100004, 53110100002, 61200020006, 62200010001, 63010010001, 70200002007, 72100001000, 431110000299

Offset: 1

Marc Morgenegg, Nov 21 2022

Other terms include: 640010100011, 722000010001, 722000010010, 730100010001, 730100010010, 802000002008, 802000002080, 821000001000, 6500011000111, 7400100100011, 7400100100101, 8301000010001, 9020000002009, 9020000002090, 9020000002900, 9210000001000.
This sequence is finite. The last term starts with 99999999898... and has 89 digits.
This sequence is for base b=10. For each base b > 2, the last term of the corresponding sequence has b^2 - b - 1 digits.
For b > 2, the final term of the sequence equals b^((b-1)^2 - 2) - (b-1)*b^((b-1)*b - 1) + ((b^(b-1) - 1)/(b-1)) * b^(b-1) * ((b-1)*b^(b*(b-1)) - b^((b-1)^2 + 1) + 1)/(b^(b-1) - 1)^2. The base-b expansion of this number is the concatenation of b-2 digits b-1, 1 digit b-2, 1 digit b-1, b-3 digits b-2, and b-1 digits k for each k in b-3..0. This is equivalent to taking a string of digits consisting of b-1 copies of every valid base-b digit (0..b-1), sorting its digits in descending order, removing one of the digits b-2, and then swapping the positions of the last digit b-1 and the first digit b-2. (Thus, for b=10, the base-10 expansion of the final term is the concatenation of eight 9's, one 8, one 9, seven 8's, nine 7's, nine 6's, ..., nine 1's, and nine 0's.) - Jon E. Schoenfield, Nov 21 2022

			63010010001 is a term: we have six 0's, three 1's, one 3 and one 6 as digits in the term, visualized as follows:
  Digits: 0123456789
   term:  63010010001.
Note that this example also shows, starting from the 11th digit, there is no more representation of the frequency of that digit, because only the first b digits of its base-b expansion count the occurrences of the corresponding digit. In this case, the last digit, 1, is the 11th.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Michael S. Branicky, Python program

Cf. A046043, A138480.

Python
```
# see linked program
```

a(14)-a(19) inserted and a(25) from Michael S. Branicky, Nov 21 2022

A358711 Autobiographical numbers: let the k-th digit count the k-th nonnegative integer (A001477(k)) (possibly overlapping) occurrences in the term.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 9321000001201000, 94201000012110000

Offset: 1

Marc Morgenegg, Nov 28 2022

The k-th digit must count the k-th nonnegative integer (A001477(k)) appearances in the term.
This sequence is in base b=10. The number of appearances of any integer is always less than b in a term. E.g., the integer '0' can appear at most 9 times in a term.
There are no further terms. This was verified with a computer search of all (permutations of) partitions of d = 1..90 using up to 9 of any digit 0..9 and all (permutations of) "completions" of the remaining d-10 digits consistent with these digit counts. It was verified in each of the two cases for counting appearances: without overlaps (1111 has 2 11's) and with overlaps allowed (1111 has 3 11's). - Michael S. Branicky, Dec 02 2022

			6401101000310 is a term: we have six 0's, four 1's, zero 2's, one 3, one 4, one 6, three 10's and one 11 as integers in the term, visualized as follows:
  Digits(k): 0 1 2 3 4 5 6 7 8 9 10 11 12 (also the Integers(k))
  term:      6 4 0 1 1 0 1 0 0 0  3  1  0
Note that overlapping integers are counted as well: e.g., 110 is one 11, one 10. 111 is two 11's.

Michael S. Branicky, Python program

Cf. A046043, A138480, A001477.

a(8) inserted and a(10)-a(15) by Michael S. Branicky, Nov 28 2022

cp's OEIS Frontend

A138481 Mutually-praising pairs excluding autobiographical numbers A046043. Version 1: both numbers in a pair have at most 10 digits.

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A138480 Autobiographical numbers: the first digit specifies how many 0's in the number, the next digit specifies how many 1's, etc. This version is not limited to 10 digits.

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A108551 Self-descriptive numbers in various bases represented in base 10.

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A138482 Mutually-praising pairs excluding autobiographical numbers A138480. Version 2: numbers may have more than 10 digits.

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A109167 a(n) = least nonnegative integer such that n appears a(n) times.

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Python

A359049 Autobiographical numbers k whose decimal digits are a concatenation count(0), count(1), ..., count(m) for some m, where count(j) is the number of (possibly overlapping) occurrences of j within the digits of k itself.

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A226195 Numbers n = x0 x1 x2...x9 such that xi is the number of digits greater than i in n.

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Maple

A234512 Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.

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Maple

A358538 Autobiographical numbers: the first digit of the term counts how many 0's the term in total has, the second how many 1's etc. up to the last digit but no more than b-1, where b is the base of the sequence. This sequence is in base b=10.

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