cp's OEIS Frontend

Note that the numbers 3,5,6,11, etc. do not appear.

Let the string (n1,n2,n3...n9) represent the number of times the digits 1,2,3...9 have appeared so far in all terms up to and including a(n). If a(n) has decimal expansion di dj ... dr then a(n+1) = ni*di + nj*dj + ... + nr*dr.

Let S(r) denote the analogous sequence obtained with initial term a(1) = r. The present sequence is S(1), and S{2} = 2, 2, 4, 8, 8, 16, 7, 7, 14, 14, 19, ...

S{11} = 11, 2, 2, 4, 4, 8, 8, 16, 9, 9, ... is the same as S{1} except at the start where 11 replaces 1,1 (there is no distinction between 1,1 and 11 in the generation of terms).

Scott R. Shannon's plot has an unusual paint-sprayer appearance. - N. J. A. Sloane, Jan 02 2020

From Michael De Vlieger, Dec 31 2020: (Start)

The "paint sprayer" streams k relate to the set D of nonzero distinct digits d in m, the term a(n-1) that precedes a(n) in the stream.

We may compactify the set D as k = Sum_{d in D} 2^(d - 1) and thus label each stream k. Since there are b nonzero base-b digits and since we cannot have a null D except in the case of m=0 (which is not in the sequence), there are thus N = 2^(b-1)-1 streams. For the decimal case herein, we have N = 2^9-1 = 511 possible streams. The base b=6 case thus has N = 2^5-1 or 31 streams.

Let c_d(n) be the number of digits d in a(n) and let s_d(n) be the partial sums of c_d(n) for 1 <= n. We may redefine a(n) = Sum_{d in D} s_d(n). This explains the "curve" of the streams in the plot and their seemingly random crossings.

The lower bound of a(n) relates to the stream k = 1, stemming from progenitor m whose digits are limited to 1s and possibly some 0s. For n > 96, the lower bound of a(n) = s_1(n), the total number of 1s in the sequence at index n.

The upper bound of a(n) relates to the sum of all s_d(n), however, we do not see a term m that instigates a stream k = 511 until we have "pandigital" m. For the purposes of this sequence, we call a number that has at least one copy of all nonzero digits d in base b.

The first instance of pandigital m is a(3994834)=127643598. We see that the following term 137801427 sets a record and is the first term in the stream k = 511. In base b=6, the smallest pandigital m is a_6(450)=2050, which is written "13254" in base 6.

We can thus identify the first term m that instigates all streams 1 <= k <= 511.

For a(n) = m with 1 <= n <= 2^22, we see m repeated at most 4 times. An example of m appearing 4 times is m = 50 at n in {43, 47, 59, 60}. We can certify 10 such occurrences of m repeated 4 times in the stated range.

a(n) = n for n in {1, 8, 573} and 1 <= n <= 2^22. Are there any more instances of a(n) = n? We note that s_1(n) > n for 877824 <= n <= 2007412. Therefore it is possible to have a(n) = n outside of that range.

(End)

This sequence is a version of the "paint sprayer" sequence A279818 that engages all decimal digits, including d = 0. For each digit d in a(n-1), a(n) records the sum of all same digits d observed in terms a(i); i = 1..n-1, summed over every d in a(n-1), wherein each occurrence of digit d = 0 in a(n-1) and prior terms is counted as 1 instead of 0.

The sums form 2^10 or 1024 trajectories similar to those in A279818, which exhibits 2^9 trajectories, since digit 0 has no effect on the sums in that sequence.

Let signature S be the set of digits in the number m; for reference we may order the digits least to greatest. For example, for m = 90210, S(90210) = {0, 1, 2, 9}. Signature S may be compactified via the sum of 2^k for k in S. Thus, for example, S(90210) compactified is 2^0 + 2^1 + 2^2 + 2^9 = 1+2+4+512 = 521.

Let trajectory T(S) be that associated with S. In scatterplot, the trajectories are arranged such that the smaller compactified signatures have the least "slope" while the pandigital signature 1023 has the greatest slope.

This trajectory arrangement is present and clearer in A279818, but those signatures involving 0 (odd compactified signatures) in this sequence are skewed, since 0 cannot begin a decimal number m > 0, and thereby has a smaller frequency.

Compared to A279818, the scatterplot appears to have denser trajectories.

The number 5 cannot appear in the sequence for n > 18.

First pandigital number is a(28035151) = 1053287964.

The signature S = {5} has not appeared for n < 10^8. It will first appear for some repdigit where the repeated digit is d = 5.

Definition related to that of the paint sprayer sequence A279818 so we may see similar behaviors and plots for this sequence.

From Michael De Vlieger, Dec 08 2024: (Start)

Let c(d) be the cardinality of digits d in a(k), k = 1..n-m. Then a(n) is the sum of d*c(d) across the set of (distinct) digits in a(n-m), m = 2. Then it is easy to see that zeros have no effect on the sums.

Scatterplot shows "rays" associated with distinct digits in a(n-2), therefore there are 2^(b-1) rays. Since b = 10, there are 512 rays in the plot, just as there are in A279818.

A279818 is the case regarding m = 1.

A378360(n) is the sum of (d+[d=0])*c(d) across the set of digits in a(n-m), m = 1, starting instead with a(1) = 0, so as to count zeros, and thus the scatterplot shows 1024 rays. (End)