A322182 -id:A322182 - cp's OEIS frontend

A325721 Start the sequence with a(1) = 1 and read the digits one by one from there. The sequence is always extended with the sum d + k, d being the digit read and k the number of d digits present so far in the sequence.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 1, 5, 4, 7, 6, 9, 8, 11, 10, 5, 6, 7, 2, 8, 9, 10, 4, 11, 12, 8, 3, 7, 9, 10, 11, 5, 12, 6, 11, 13, 12, 4, 13, 14, 9, 15, 6, 10, 16, 17, 18, 7, 19, 7, 8, 20, 8, 21, 9, 14, 22, 10, 11, 23, 5, 24, 12, 25, 12, 26, 13, 13, 27, 15, 14, 14, 8, 6, 15, 9, 28, 16, 29, 10, 10, 11, 30, 7, 31, 32, 12, 9, 11, 13, 11

Offset: 1

Eric Angelini and Carole Dubois, May 16 2019

			The sequence starts with a(1) = 1.
We read this 1, see that there is only one digit 1 so far in the sequence, thus k = 1; we have then d + k = (1 + 1) = 2 and this 2 becomes a(2);
We read this 2, see that there is only one digit 2 so far in the sequence, thus k = 1; we have then d + k = (2 + 1) = 3 and this 3 becomes a(3);
We read this 3, see that there is only one digit 3 so far in the sequence, thus k = 1; we have then d + k = (3 + 1) = 4 and this 4 becomes a(4);
...
We now read the first digit of a(10) = 10, see that this 1 is the 2nd digit 1 so far in the sequence, thus k = 2; we have then d + k = (1 + 2) = 3 and this 3 becomes a(11);
We now read the second digit of a(10) = 10, see that this 0 is the 1st digit 0 so far in the sequence, thus k = 1; we have then d + k = 1 and this 1 becomes a(12);
We now read the single digit of a(11) = 3; we see that this 3 is the 2nd digit 3 so far in the sequence, thus k = 2; we have then d + k = 5 and this 5 becomes a(13); etc.

Carole Dubois, Table of n, a(n) for n = 1..5001
Carole Dubois, Digit-count for this sequence and two others visible in the Xref section

Cf. A325722 where the same idea is developed, but with d*k instead of d+k. See also A308232 for the concatenation kd.

Cf. A322182.

A325722 Start the sequence with a(1) = 1 and read the digits one by one from there. The sequence is always extended with the product d*k, d being the digit read and k the number of d digits present up to the point of the d digit taken.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 16, 3, 6, 3, 12, 6, 4, 6, 18, 12, 24, 5, 24, 6, 8, 10, 16, 5, 12, 20, 30, 32, 7, 0, 8, 36, 10, 9, 14, 16, 0, 9, 0, 12, 18, 7, 0, 40, 15, 42, 10, 0, 9, 11, 24, 12, 48, 0, 18, 0, 13, 20, 14, 48, 14, 0, 28, 0, 15, 15, 32, 22, 16, 0, 0, 27, 17, 18, 24, 36, 19, 26, 40, 56, 0, 20, 64, 0, 21, 18, 28, 0, 22, 44, 48, 72

Offset: 1

Eric Angelini and Carole Dubois, May 16 2019

Every prime number > 7 appears only once and in natural order. - Davide Rotondo, Feb 08 2024

			The sequence starts with a(1) = 1.
We read this 1, see that there is only one digit 1 so far in the sequence, thus k = 1; we have then d*k = 1 and this 1 becomes a(2);
We read a(2) = 1, see that this 1 is the 2nd digit 1 so far in the sequence, thus k = 2; we have then d*k = 2 and this 2 becomes a(3);
We read a(3) = 2, see that there is only one digit 2 so far in the sequence, thus k = 1; we have then d*k = 2 and this 2 becomes a(4);
We read a(4) = 2, see that this 2 is the 2nd digit 2 so far in the sequence, thus k = 2; we have then d*k = 4 and this 4 becomes a(5);
...
We read now the first digit of a(9) = 16 and see that this 1 is the 3rd digit 1 so far in the sequence, thus k = 3; we have then d*k = 3 and this 3 becomes a(10);
We read now the second digit of a(9) = 16 and see that this 6 is the 1st digit 6 so far in the sequence, thus k = 1; we have then d*k = 6 and this 6 becomes a(11); etc.
From _Kevin Ryde_, Feb 10 2024: (Start)
Digits d from the sequence terms, their respective occurrence number k, and consequent terms a(n) = d*k, begin:
     d = 1 1 2 2 4 4 8  8  1  6  3  6  3  1  2  6  4 ...
     k = 1 2 1 2 1 2 1  2  3  1  1  2  2  4  3  3  3 ...
  d*k  = 1 2 2 4 4 8 8 16  3  6  3 12  6  4  6 18 12 ...
(End)

Carole Dubois, Table of n, a(n) for n = 1..5001
Carole Dubois, Digit-count for this sequence and two others visible in the Xref section

Cf. A325721 where the same idea is developed, but with d+k instead of d*k. See also A308232 for the concatenation kd.

Cf. A322182.

digs(x) = if (x, digits(x), [0]);
countd(listd, posd, y) = my(nb=0); for (k=1, posd, if (listd[k] == y, nb++);); nb;
lista(nn) = my(list=List(1), listd=List(1), pos=1, posd=1); for (n=1, nn, my(d = digs(list[pos])); for (i=1, #d, my(y = d[i], nb = countd(listd, posd, y)); listput(list, y*nb); my(dd = digs(y*nb)); for (j=1, #dd, listput(listd, dd[j]);); posd++;); pos++;); Vec(list); \\ Michel Marcus, Feb 09 2024

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

A325721 Start the sequence with a(1) = 1 and read the digits one by one from there. The sequence is always extended with the sum d + k, d being the digit read and k the number of d digits present so far in the sequence.

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A325722 Start the sequence with a(1) = 1 and read the digits one by one from there. The sequence is always extended with the product d*k, d being the digit read and k the number of d digits present up to the point of the d digit taken.

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PARI

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.

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Crossrefs

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MATLAB

Python