A319678 -id:A319678 - cp's OEIS frontend

A343963 a(0) = 0, and for any n > 0, the binary expansion of n has n digits and starts with the binary expansion of n, say of w digits, and in case n > w, the remaining binary digits in a(n) are those of a(n-w).

0, 1, 2, 7, 9, 22, 55, 121, 137, 310, 695, 1529, 3209, 6966, 15031, 32249, 34297, 72841, 154422, 326327, 687609, 1410553, 2956425, 6183734, 12909239, 26902009, 55936505, 116202633, 241064758, 499448503, 1033534969, 2136311289, 2203420153, 4545387657

Offset: 0

Rémy Sigrist, May 05 2021

To build the binary expansion of a(n):
- start with n indeterminate digits,
- while there are some, say m, indeterminate digits,
  replace the first of them with the binary expansion of m.
The binary plot of the sequence has locally periodic patterns.

			For n = 10:
- the binary expansion of a(10) has 10 digits, and is the concatenation of:
   - the binary expansion of 10 which is "1010",
   - the binary expansion of 10 - 4 = 6 which is "110",
   - the binary expansion of 10 - 4 - 3 = 3 which is "11",
   - the binary expansion of 10 - 4 - 3 - 2 = 1 which is "1",
   - as 10 = 4 + 3 + 2 + 1, we stop here,
- so the binary expansion of a(10) is "1010110111",
- and a(10) = 695.

Rémy Sigrist, Binary plot of the sequence for n <= 2^10

Cf. A070939, A319678.

a(n) = { if (n==0, 0, my (k=n-#binary(n)); n*2^k+a(k)) }

A070939(a(n)) = n for any n > 0.

A348111 Numbers k whose binary expansion starts with the concatenation of the binary expansions of the run lengths in binary expansion of k.

Original entry on oeis.org

0, 1, 7, 14, 28, 112, 127, 254, 509, 1016, 1018, 1792, 2033, 2037, 2039, 4066, 4072, 4075, 4078, 8132, 8135, 8150, 8156, 16256, 16300, 16313, 32513, 32528, 32576, 32601, 32607, 32626, 32639, 32767, 65027, 65087, 65153, 65202, 65248, 65253, 65255, 65534, 130307

Offset: 1

Rémy Sigrist, Oct 01 2021

We consider here that 0 has an empty binary expansion, and include it in the sequence.

This sequence is infinite as it contains A077585.

			Regarding 32607:
- the binary expansion of 32607 is "111111101011111",
- the corresponding run lengths are: 7, 1, 1, 1, 5,
- in binary: "111", "1", "1", "1", "101",
- after concatenation: "111111101",
- as "111111101011111" starts with "111111101", 32607 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A348111

Cf. A077585, A101211, A175930, A319678.

PARI
```
See Links section.
```

from itertools import groupby
def ok(n):
    if n == 0: return True
    b = bin(n)[2:]
    c = "".join(bin(len(list(g)))[2:] for k, g in groupby(b))
    return b.startswith(c)
print(list(filter(ok, range(2**17)))) # Michael S. Branicky, Oct 02 2021

A357118 Numbers such that the first digit is the number of digits and the second digit is the number of distinct digits.

Original entry on oeis.org

322, 323, 4222, 4224, 4242, 4244, 4300, 4303, 4304, 4311, 4313, 4314, 4322, 4323, 4324, 4330, 4331, 4332, 4335, 4336, 4337, 4338, 4339, 4340, 4341, 4342, 4345, 4346, 4347, 4348, 4349, 4353, 4354, 4355, 4363, 4364, 4366, 4373, 4374, 4377, 4383, 4384, 4388, 4393, 4394, 4399

Offset: 1

Marc Morgenegg, Oct 17 2022

By definition, each term must be at least a two-digit number.
The sequence is finite. The last term is 989765432.
There are 11041830 terms. - Michael S. Branicky, Oct 20 2022

			322 is a term because the first digit is the number of digits = 3, and the second digit counts the different digits {2,3} = 2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A319678.

def ok(n):
    s = str(n)
    if len(s) < 2 or len(s) > 9: return False
    return len(s) == int(s[0]) and len(set(s)) == int(s[1])
print([k for k in range(4394) if ok(k)]) # Michael S. Branicky, Oct 17 2022

cp's OEIS Frontend

A343963 a(0) = 0, and for any n > 0, the binary expansion of n has n digits and starts with the binary expansion of n, say of w digits, and in case n > w, the remaining binary digits in a(n) are those of a(n-w).

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A348111 Numbers k whose binary expansion starts with the concatenation of the binary expansions of the run lengths in binary expansion of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A357118 Numbers such that the first digit is the number of digits and the second digit is the number of distinct digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python