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, 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

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

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Rémy Sigrist, PARI program for A309079

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.

a(A164894(k)) = A000225(k) for any k > 0.

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

Cf. A000225, A143789, A164894.

Sequence in context: A117607 A215092 A345018 * A194551 A212644 A351285

Adjacent sequences: A309076 A309077 A309078 * A309080 A309081 A309082

Keyword

nonn,base

Author

Rémy Sigrist, Jul 11 2019

Status

approved