cp's OEIS Frontend

We begin with the empty sum 0.

This sequence also counts zeros in the decimal expansion of a number.

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.

a(n+1) is k(n)*sopfr(a(n)), where k(n) is the number of times (up to and including a(n)) that a term having the same sopfr as a(n) has occurred in the sequence so far.

The smallest possible initial term is a(1)=2, since 2 is the smallest number having a prime divisor. Not every integer appears; for example, 3 cannot occur unless chosen as a(1) (in which case 2 cannot appear).

The primes do not appear in their natural order (e.g., 31 precedes 29). If the lesser of twin primes p is a(k) and the greater twin has not already occurred, then a(k+2) is the greater twin. Whereas if a composite number appears, it may appear more than once, a prime > 2, if it appears, can appear once only. Open question: Do all primes (except for 3) eventually appear?

a(n+1) = k(n)*q(a(n)), where k(n) is the number of times (up to and including a(n)) that a term having the same q-value as a(n) has occurred in the sequence so far.

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)

For all n, we have c_1 = (n-1) since 1 | n for all nonzero n.

There are 2 means by which we set c_p = 1 for p prime. Let m = a(n), the progenitor of a(n+1). The first is directly via Sum_{d | a(n)} c_d = p. The second is indirectly, through Sum_{d | a(n)} c_d = m, and novel p | m. An example of the first is the appearance of a(3) = 2, and the second, a(6) = 6 is divisible by 3, a prime not yet appearing in the sequence.

For a(n) = p novel, a(n+1) = n, since we have (n-1) instances of d = 1 and 1 instance of p. Subsequent appearances of p have a(n+1) exceed n. Primes may appear more than once.

The two instances of 1 that begin the sequence yield a(3) = 2. The terms a(2) and a(3) are the only instances of a(n) < n. This is because only register c_1 > 0 for those terms, and in all other terms, we have the progenitor term a(n) > 1. Indeed, 1 can never again arise, because the only way we have Sum_{d | a(n)} c_d = 1 is when novel a(n) = 1 where d = 1 has only appeared once before. Clearly 1 has already appeared twice for n >= 2. No other number has 1 divisor, therefore there are only 2 occasions of 1 in the sequence, i.e., a(1) = a(2) = 1.

Generally for n > 3, m < n does not appear as a term. This implies the lower bound a(n) = n for n > 3. We have shown that a(n) = n results from certain prime progenitors m.

The striations seen in the scatterplot relate to the lesser prime divisors of progenitors a(n) = m -> a(n+1).

In other words, a(n+1) is the sum, taken over each distinct d in a(n), of the number of pairs (d,d) counted from different occurrences of d in a(j), 0 <= j <= n.

Every term is the sum of one or more triangular numbers (A000217). If no digit in a(n) has occurred before, a(n+1) = 0 (there are 7 such terms). Some numbers (0,1,3,4,6,10,...) occur multiple times, while others (2,5,8,11,...) never occur.

a(28844) = 2417583609 is the first pandigital term. Similar to A279818.