A336533 Lexicographically earliest sequence of positive terms such that for any n > 0, n = Sum_{k >= 0} b(k)*a(k+1) where Sum_{k >= 0} b(k)*2^k is the binary expansion of a(n).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 23, 26, 27, 29, 30, 31, 47, 55, 58, 59, 61, 62, 63, 93, 94, 95, 111, 119, 122, 123, 125, 126, 127, 191, 221, 222, 223, 239, 247, 250, 251, 253, 254, 255, 382, 383, 447, 477, 478, 479, 495, 503, 506, 507, 509, 510, 511, 767

Offset: 1

Author

Rémy Sigrist, Sep 26 2020

Keywords

• nonn
• base

Comments

In other words, the binary expansion of the n-th term encodes a partition of n into distinct terms of the sequence.

This sequence is complete (as any integer can be written as a sum of distinct terms of this sequence).

Examples

			The first terms, alongside their binary representation and the corresponding partition of n, are:
  n   a(n)  bin(a(n))  Partition of n
  --  ----  ---------  -------------------------
   1     1          1  a(1)
   2     2         10  a(2)
   3     3         11  a(2) + a(1)
   4     5        101  a(3) + a(1)
   5     6        110  a(3) + a(2)
   6     7        111  a(3) + a(2) + a(1)
   7    10       1010  a(4) + a(2)
   8    11       1011  a(4) + a(2) + a(1)
   9    13       1101  a(4) + a(3) + a(1)
  10    14       1110  a(4) + a(3) + a(2)
  11    15       1111  a(4) + a(3) + a(2) + a(1)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Binary plot of the first 1000 terms
• Rémy Sigrist, PARI program for A336533
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A029931, A133457.

Programs

PARI

See Links section.

Formula

a(Sum_{k = 1..n} a(k)) = 2^n - 1 for any n > 0.