A336514 -id:A336514 - cp's OEIS frontend

A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 11, 13, 2, 2, 3, 3, 4, 2, 4, 3, 5, 2, 5, 3, 6, 2, 6, 3, 7, 2, 7, 3, 8, 2, 8, 3, 9, 2, 9, 3, 10, 15, 4, 4, 5, 5, 6, 4, 6, 5, 7, 4, 7, 5, 8, 4, 8, 5, 9, 4, 9, 5, 10, 17, 6, 6, 7, 7, 8, 6

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(19)=10, the terms are identical to A248034. The branches (distinct lines of terms indicating the largest digit of the preceding term) can be labeled by the counter digit (shown in the scatter plot). From 9 to 1 the branches gradually get fragmented. Below digit 5 it is harder to disentangle the branches in some places. A repeating pattern also appears (shown in the inset of the scatter plot).

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves
Bence Bernáth, Scatter plot for n=0..500000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^6.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5, with a color function indicating largest digit of preceding term, where black = 0, red = 1, orange = 2, ..., magenta = 9.

Cf. A248034, A249009, A356348, A336514.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==max((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

nn = 79; s[] = 0; a[0] = 0; Do[(Set[d, Max[#]]; Map[s[#1] += #2 & @@ # &, Tally[#] ]) &@ IntegerDigits[a[n - 1]]; a[n] = s[d], {n, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger, Dec 28 2022 *)

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
     sequence.append(seq_for_digits.count(max(list(map(int,list(str(sequence[-1])))))))
     seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A336515 a(1) = 1, and for any n > 0, a(n+1) is the number of k in the range 1..n such that the binary representation of a(k) appears as a substring in the binary representation of a(n).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 4, 6, 6, 7, 5, 5, 6, 8, 7, 6, 9, 7, 7, 8, 8, 9, 8, 10, 7, 9, 9, 10, 8, 11, 9, 11, 10, 9, 12, 13, 13, 14, 15, 10, 10, 11, 11, 12, 14, 16, 12, 15, 11, 13, 15, 12, 16, 13, 16, 14, 17, 12, 17, 13, 17, 14, 18, 13, 18, 14, 19, 15, 13, 19, 16

Offset: 1

Rémy Sigrist, Jul 24 2020

This sequence is unbounded.

			The first terms, alongside their binary representation and the corresponding k's, are:
  n   a(n)  bin(a(n))  k's
  --  ----  ---------  -------------
   1     1          1  N/A
   2     1          1  {1}
   3     2         10  {1, 2}
   4     3         11  {1, 2, 3}
   5     3         11  {1, 2, 4}
   6     4        100  {1, 2, 4, 5}
   7     4        100  {1, 2, 3, 6}
   8     5        101  {1, 2, 3, 6, 7}
   9     4        100  {1, 2, 3, 8}
  10     6        110  {1, 2, 3, 6, 7, 9}

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Density plot of the first 100000000 terms
Rémy Sigrist, PARI program for A336515

Cf. A336514 (decimal variant).

PARI
```
See Links section.
```

A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10, 8, 8, 9, 8, 10, 9, 9, 10, 10, 11, 22, 12, 23, 14

Offset: 0

Bence Bernáth, Dec 08 2022

Up to a(103)=12, the terms are identical to A248034.

Bence Bernáth, Table of n, a(n) for n = 0..10000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358851.

length_seq=150;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==min((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

sequence=[0]
length=150
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
   sequence.append(seq_for_digits.count(min(list(map(int,list(str(sequence[-1])))))))
   seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 4, 10, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 6, 6, 7, 6, 8, 6, 9, 6, 10, 7, 7, 8, 7, 9, 7, 10

Offset: 0

Bence Bernáth, Dec 12 2022

The mode is the most frequently occurring value among the digits of a(n). When there are multiple values occurring equally frequently, the mode is the smallest of those values.

Up to a(464)=110, the terms are identical to A358967.

Bence Bernáth, Table of n, a(n) for n = 0..20000
Eric Angelini, Digit-counters updating themselves

Cf. A248034, A249009, A356348, A336514, A358967, A358851, A322182.

length_seq=470;
sequence(1)=0;
seq_for_digits=(num2str(sequence(1))-'0');
for i1=1:1:length_seq
     sequence(i1+1)=sum(seq_for_digits==mode((num2str(sequence(i1))-'0'))');
     seq_for_digits=[seq_for_digits, num2str(sequence(i1+1))-'0'];
end

import statistics as stat
sequence=[0]
length=470
seq_for_digits=list(map(int, list(str(sequence[0]))))
for ii in range(length):
    sequence.append(seq_for_digits.count(stat.mode(list(map(int, list(str(sequence[-1])))))))
    seq_for_digits.extend(list(map(int, list(str(sequence[-1])))))

cp's OEIS Frontend

A358851 a(n+1) is the number of occurrences of the largest digit of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

Mathematica

Python

A336515 a(1) = 1, and for any n > 0, a(n+1) is the number of k in the range 1..n such that the binary representation of a(k) appears as a substring in the binary representation of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A358967 a(n+1) gives the number of occurrences of the smallest digit of a(n) so far, up to and including a(n), with a(0)=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

Python

A359031 a(n+1) gives the number of occurrences of the mode of the digits of a(n) among all the digits of [a(0), a(1), ..., a(n)], with a(0)=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

Python