A046043 -id:A046043 - cp's OEIS frontend

A256104 Differential autobiographical numbers: number n = x0 x1 x2 ... x9 such that xi is the number of pairs (xj, xk), j different from k, where |xj - xk| = i.

20404, 31330

Offset: 1

Michel Lagneau, Mar 14 2015

The first digit specifies how many |xj - xk| = 0 in the number, the next digit specifies how many |xj - xk| = 1, etc.

			31330 is in the sequence because:
|x0 - x2| = 0, |x0 - x3| = 0 and |x2 - x3| = 0 => x0 = 3;
|x1 - x3| = 1 => x1 = 1;
|x0 - x1| = 2, |x1 - x2| = 2 and |x1 - x3| = 2 => x2 = 3;
|x0 - x4| = 3, |x2 - x4| = 3 and |x3 - x4| = 3 => x2 = 3;
|xj - xk| = 4 does not occur for all j and k => x4 = 0.

Cf. A046043, A234512.

for n from  10 to 10^10 do:
  x:=convert(n,base,10):n0:=nops(x):T:=array(0..9):
    for a from 0 to 9 do:
    T[a]:=0:
    od:
     for i from 0 to 9 do:
      for j from 1 to n0-1 do:
       for k from j+1 to n0 do:
       if abs(x[j]-x[k])= i
       then
       T[i]:=T[i]+1:
       else
       fi:
     od:
    od:
   od:
    s:=sum('T[m]*10^(n0-m-1)', 'm'=0..9):
    if s=n then print(n) else fi:od:

A260387 Numbers n = d_0d_1...d_n (n < 10) such that d_i is the number of digits equal to i in n (base b), where b is less than 10.

Original entry on oeis.org

12, 13, 320, 3201, 72200, 89000, 132110, 345000, 643000, 2320200, 3121300, 10103111, 11300130, 42430000, 51340000, 64030000, 72300000, 86300000, 125102000, 130213000, 211220001, 220101111, 323111000, 431130000, 614110000, 667000000, 2153100000, 2521002000, 3021211100

Offset: 1

Pieter Post, Jul 24 2015

The only terms having the same number of digits as the base are 13, 10103111, 211220001 and 220101111. For example, 13 is 1101_2,  which has 1 zero and 3 ones.
The least term with 10 digits that describes itself is 2153100000.
2153100000 is 104233022322_7, so it has 2 zeros, 1 one, 5 twos, 3 threes, 1 four, 0 fives, 0 sixes, 0 sevens, 0 eights and 0 nines in base 7.

			12 = 110_3, which has 1 zero and 2 ones.
13 = 1101_2, which has 1 zero and 3 ones.
320 = 11000_4, which has 3 zeros, 2 ones and 0 twos.
3201 = 100301_5, which has 3 zeros, 2 ones, 0 twos and 1 three.
72200 = 10200001002_3
89000 = 10101101110101000_2
132110 = 13211420_5
345000 = 122112020210_3
643000 1012200000211_3
42430000 = 2201312320300_4
51340000 = 3003312023200_4
64030000 = 3310100110300_4
72300000 = 122002100000_5
86300000 = 20000101111100022_3
431130000 = 110440340120_6
614110000 = 2224203010000_5
667000000 = 1201111002002222201_3
2153100000 = 104233022322_7

Cf. A001155, A001387, A005150, A046043, A138480.

a(10)-a(13), a(19)-a(23), a(28)-a(29) added by Giovanni Resta, Jul 26 2015

A282535 a(n) is the maximum number of "describing"-steps for an n-chain before entering a loop.

Original entry on oeis.org

3, 4, 7, 4, 7, 7, 7, 6, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 4

Felix Fröhlich, Feb 17 2017

Conjecture: The size of terms of this sequence is unbounded (cf. Marichal, 2007, Corollary 5).

Jean-Luc Marichal, On perfect, amicable, and sociable chains, arXiv:0708.1491 [math.CO], 2007.

Cf. A046043, A108551.

pad(d, n) = while(#d != n, d = concat([0], d)); d;
say(d, n) = vector(n, k, sum(j=1, #d, d[j] == (k-1)));
isok(v, n) = my(vs = vecsort(v,,8)); (#vs > 1) && (#vs Michel Marcus, Feb 25 2017

Name edited by Michel Marcus, Feb 26 2017

A380985 Numbers whose k-th digit indicates the number of digits which occur k times.

Original entry on oeis.org

1, 20, 110, 2100, 20100, 200100, 2000100, 20000100, 200000100, 2000000100, 20000000100, 200000000100, 2000000000100, 20000000000100, 200000000000100, 2000000000000100, 20000000000000100, 200000000000000100, 2000000000000000100, 20000000000000000100

Offset: 1

Leo Crabbe, Feb 11 2025

The most significant digit is digit position k=1, meaning that it counts how many numbers appear 1 time.

			110 is a term because 1 number appears once (0), 1 number appears twice (1) and 0 numbers appear 3 times.

Cf. A046043.

from collections import Counter
def ok(n):
    d = list(map(int, str(n)))
    c = Counter(Counter(d).values())
    return all(dk == c[k] for k, dk in enumerate(d, 1)) # Michael S. Branicky, Feb 18 2025

a(n) = 2*10^(n-1) + 100 for n >= 4. - Andrew Howroyd, Feb 22 2025

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A256104 Differential autobiographical numbers: number n = x0 x1 x2 ... x9 such that xi is the number of pairs (xj, xk), j different from k, where |xj - xk| = i.

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A260387 Numbers n = d_0d_1...d_n (n < 10) such that d_i is the number of digits equal to i in n (base b), where b is less than 10.

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A282535 a(n) is the maximum number of "describing"-steps for an n-chain before entering a loop.

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A380985 Numbers whose k-th digit indicates the number of digits which occur k times.

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