A340619 n appears A006519(n) times.

1, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 19, 20, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 24, 24, 24, 24, 24, 25, 26, 26

Offset

1,2

Comments

This sequence has similarities with the Cantor staircase function.

This sequence can be seen as an irregular table where the n-th row contains A006519(n) times the value n.

For any k > 1, the set of points { (n, a(n)), n = 1..A006520(2^k-1) } is symmetric; for example, for k = 3, we have the following configuration:

a(n)

^

| *

| **

| *

| ****

| *

| **

|*

+-------------> n

Links

Rémy Sigrist, Table of n, a(n) for n = 1..11264

Wikipedia, Cantor function

Formula

a(A006520(n)) = n.

a(A006520(n)+1) = n+1.

a(n) + a(A006520(2^k-1) + 1 - n) = 2^k for any k > 0 and n = 1..A006520(2^k-1).

a(n) = 2^k + (a(r) if r>0), where k such that k*2^(k-1) < n <= (k+1)*2^k and r = n - (k+2)*2^(k-1). - Kevin Ryde, Jan 18 2021

Example

The first rows, alongside A006519(n), are:
    n | n-th row               | A006519(n)
   ---+------------------------+-----------
    1 | 1                      |          1
    2 | 2, 2                   |          2
    3 | 3                      |          1
    4 | 4, 4, 4, 4             |          4
    5 | 5                      |          1
    6 | 6, 6                   |          2
    7 | 7                      |          1
    8 | 8, 8, 8, 8, 8, 8, 8, 8 |          8
    9 | 9                      |          1
   10 | 10, 10                 |          2

Mathematica

A340619[n_] := Array[n &, Table[BitAnd[BitNot[i - 1], i], {i, 1, n}][[n]]];
Table[A340619[n], {n, 1, 26}] // Flatten (* Robert P. P. McKone, Jan 19 2021 *)

Prog

(PARI) concat(apply(v -> vector(2^valuation(v, 2), k, v), [1..26]))
(PARI) a(n) = my(ret=0); forstep(k=logint(n, 2), 0, -1, if(n > k<<(k-1), ret+=1<<k; n-=(k+2)<<(k-1))); ret; \\ Kevin Ryde, Jan 18 2021

Crossrefs

See A061392 and A340500 for similar sequences.

Cf. A006519, A006520, A046699.

Sequence in context: A087180 A029121 A161205 * A334800 A228942 A164531

Adjacent sequences: A340616 A340617 A340618 * A340620 A340621 A340622

Keyword

nonn,easy,tabf

Author

Rémy Sigrist, Jan 13 2021

Status

approved