A322000 Nonnegative integers, sorted by increasing value of A028897(n) = Sum d[i]*2^i for n = Sum d[i]*10^i, then value of n.

0, 1, 2, 10, 3, 11, 4, 12, 20, 100, 5, 13, 21, 101, 6, 14, 22, 30, 102, 110, 7, 15, 23, 31, 103, 111, 8, 16, 24, 32, 40, 104, 112, 120, 200, 1000, 9, 17, 25, 33, 41, 105, 113, 121, 201, 1001, 18, 26, 34, 42, 50, 106, 114, 122, 130, 202, 210, 1002, 1010, 19, 27

Offset: 0

Author

M. F. Hasler, Feb 13 2019

Keywords

• nonn
• base
• look

Comments

A028897(n) is the result of using the decimal digits of n, but weighting their position as in base 2. For sake of brevity we refer to this as the b-value of n in the sequel. This idea is found on the website given in links under the name "decibinary numbers".

The b-values increment by 1 at indices (of "records") 1, 2, 4, 6, 10, 14, 20, 26, 36, ... Prefixing an initial 0, the gaps between these, equal to the number of occurrences of a given b-value (0, 1, 2, ...), are 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ... = A072170(n,10). In this sequence each of (1, 2, 4, 6, 10, 13, 18, ...) is repeated twice.

Examples

			The first terms of the sequence are as follows: (b = A028897)
  n | 0 | 1 | 2 | 10 | 3 | 11 | 4 | 12 | 20 | 100 | 5 | 13 | 21 | 101 | ...
----+---+---+---+----+---+----+---+----+----+-----+---+----+----+-----+-----
b(n)| 0 | 1 | 2 |  2 | 3 |  3 | 4 |  4 |  4 |  4  | 5 |  5 |  5 |  5  | ...
For example, b(345) = 3*2^2 + 4*2 + 5 = 25.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Robin C. Yu, Decibinary Numbers, on Hackerrank.com.

Crossrefs

Cf. A028897, A072170 (see comments).

Programs

Maple

N:= 30: # for all numbers with A028897(n) <= N
L:= {seq([i,i],i=0..9)}: Agenda:= {seq([i,i],i=1..9)}:
extend:= proc(p) local x;  op(select(t -> t[2]<=N, [seq([10*p[1]+x, 2*p[2]+x],x=0..9)])); end proc:
sorter:= proc(p1,p2) if p1[2] <> p2[2] then p1[2] < p2[2] else p1[1] < p2[1] fi end proc:
while Agenda <> {} do
  Agenda:= map(extend, Agenda);
  L:= L union Agenda;
od:
L:= sort( convert(L,list),sorter):
map(t -> t[1], L); # Robert Israel, Feb 24 2019

PARI

my(A028897(n)=fromdigits(digits(n),2),S=[]);for(k=1,2^10,(t=A028897(k))>9||S=setunion(S,[[t,k]]));apply(t->t[2],S)