

A309079


For any n > 0: consider the strictly increasing finite sequences of integers whose concatenation of terms, in binary and without leading zeros, equals that of n; a(n) is the minimal sum of the terms of such a finite sequence.


1



1, 2, 3, 4, 5, 3, 4, 8, 9, 10, 5, 5, 6, 7, 8, 16, 17, 18, 19, 6, 7, 8, 9, 9, 10, 11, 6, 7, 8, 9, 10, 32, 33, 34, 35, 36, 9, 10, 11, 10, 11, 12, 13, 14, 15, 11, 12, 17, 18, 19, 20, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 64, 65, 66, 67, 68, 69, 70, 71, 12
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OFFSET

1,2


COMMENTS

Any integer appear in the sequence:
 for any m > 0 with binary expansion Sum_{k >= 0} b_k * 2^k,
 let n = (Sum_{k >= 0} b_k * 2^Sum_{j >= k} ((1+j) * b_j))/2,
 then a(n) = m,
 for example (in binary): a("1101000") = "1" + "10" + "1000" = "1011".


LINKS



FORMULA

a(n) <= n with equality iff n is a power of two or the binary concatenation of 2^k and m for some k >= 0 and m <= 2^k.
a(2*n) <= 2*a(n).
a(2*n + 1) <= 2*a(n) + 1.


EXAMPLE

The first terms, alongside the corresponding finite sequences, are:
n a(n) bin(n) bin(seq)
   
1 1 1 (1)
2 2 10 (10)
3 3 11 (11)
4 4 100 (100)
5 5 101 (101)
6 3 110 (1,10)
7 4 111 (1,11)
8 8 1000 (1000)
9 9 1001 (1001)
10 10 1010 (1010)
11 5 1011 (10,11)
12 5 1100 (1,100)
13 6 1101 (1,101)
14 7 1110 (1,110)
15 8 1111 (1,111)
16 16 10000 (10000)
17 17 10001 (10001)
18 18 10010 (10010)
19 19 10011 (10011)
20 6 10100 (10,100)
21 7 10101 (10,101)


PROG

(PARI) See Links section.


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



