OFFSET
1,3
COMMENTS
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)
LINKS
Scott R. Shannon, Table of n, a(n) for n = 1..20000
Michael De Vlieger, Explanation of the streams evident in the plot of this sequence.
Michael De Vlieger, Log log plot of a(n) for n = 1..4000000 showing records in red, local minima in blue, and indicating the first term of each stream k= 1..511 in gold.
Michael De Vlieger, Log log plot of s_d(n) for n = 1..2^18 using a color function to indicate d where d=1 is red, d=9 is purple, and d=0 is black.
Michael De Vlieger, Example of progenitors a(n-1) that produce a(n) that pertain to digit signature k=138 (i.e., a(n-1) has digits 1, 2, 4, and 8 and any number of zeros). a(n-1) is blue, a(n) in red.
Michael De Vlieger, Plot of a_6(n) for n = 1..6^7 (base 6 version of this sequence for comparison, exhibiting N = 2^5-1 = 31 streams.
Michael De Vlieger, Table of the least term m of a given digit signature k, or the first term in stream k.
Michael De Vlieger, Lineages k represented among the records in this sequence for n = 1..2^22.
Scott R. Shannon, Plot of a(n) for n = 1..1000000.
EXAMPLE
a(2)=1, the sum of digits of a(1); a(3)=1+1=2, the sum of digits of a(1) and a(2), the only terms so far to contain digit 1; a(4)=2, the sum of digits of a(3), the only term so far to contain digit 2; a(5)=2+2=4, since 2 is the only digit appearing in a(4) and a(3); a(6)=4, a(7)=8; etc.
MATHEMATICA
Block[{a = {1}, s}, Array[Set[s[#], 0] &, 9]; Do[(MapIndexed[AddTo[s[First[#2]], #1] &, #]; AppendTo[a, Total@ Map[# s[#] &, Position[#, _?(# > 0 &)][[All, 1]] ]]) &@ Most@ DigitCount@ Last@ a, 69]; a ] (* Michael De Vlieger, Dec 31 2020 *)
PROG
(PARI) lista(nn) = {my(a=1, vsd = vector(9)); print1(a, ", "); for (n = 2, nn, my(s = sumdigits(a)); my(dd = Set(digits(a))); my(rd = digits(a)); a = s + sum(k=1, #dd, if (dd[k], vsd[dd[k]])); print1(a, ", "); for (k=1, #rd, if (rd[k], vsd[rd[k]] += rd[k]); ); ); } \\ Michel Marcus, Nov 06 2019
CROSSREFS
KEYWORD
AUTHOR
David James Sycamore, Dec 19 2016
EXTENSIONS
Partially edited by N. J. A. Sloane, Sep 03 2021
STATUS
approved