A378360 - OEIS

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.

Michael De Vlieger, Log log scatterplot of c(d, n), n = 1..10^5, d = 0..9, with a color function showing d = 0 in black, d = 1 in red, d = 2 in orange, ..., d = 9 in purple.

Michael De Vlieger, First emergence of digit signatures in A378360 and for versions of this sequence in bases b = 2..9. Includes remarks for each base, proof that a(n) = 5 is impossible for n > 18 in decimal, and related code.

Let c(d) be the number of digits d in the sequence a(1..n-1), abbreviating the dyadic function for concision:

a(2) = 1 since 0+[0==0]*c(0) = 1*1 = 1.

a(3) = 1 since 1+[1==0]*c(1) = 1*1 = 1.

a(4) = 2 since 1*c(1) = 1*2 = 2.

a(5) = 2 since 2*c(2) = 2*1 = 2.

a(6) = 4 since 2*c(2) = 2*2 = 4.

...

a(10) = 16 since a(9) = 8, and 8*c(8) = 8*2 = 16.

a(11) = 9 since 1*c(1)+6*c(6) = 1*3+6*1 = 9.

...

a(19) = 14 since a(18) = 53, and 5*c(5)+3*c(3) = 5*1+3*3 = 14.

a(20) = 22 since 1*c(1)+4*c(4) = 1*6+4*4 = 22.

a(21) = 10 since 2*c(2) = 2*5 = 10, (not 2*c(2)+2*c(2) = 20), etc.

nn = 120; a[1] = j = 0; c[_] := 0;

Do[k = Total@ Map[(#1 + Boole[#1 == 0])*(c[#1] += #2) & @@ # &,

Tally@ IntegerDigits[j] ];

Set[{a[n], j}, {k, k}],

{n, 2, nn}]; Array[a, nn]