A361838 a(n) is the number of 2s in the binary hereditary representation of 2n.

1, 2, 3, 2, 3, 4, 5, 3, 4, 5, 6, 5, 6, 7, 8, 3, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 8, 9, 10, 11, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 12, 13, 14, 15, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7

Offset: 1

Author

Jodi Spitz, Mar 26 2023

Keywords

• nonn
• base
• hear

Comments

See comments on A266201 for the definition of hereditary representation.

Examples

			A table of n, the binary hereditary representation of 2n, and the number of 2s in the representation:
 n | hereditary rep. of 2n   | number of 2s
---+-------------------------+--------------
 1 | 2                       |      1
 2 | 2^2                     |      2
 3 | 2^2+2                   |      3
 4 | 2^(2+1)                 |      2
 5 | 2^(2+1)+2               |      3
 6 | 2^(2+1)+2^2             |      4
 7 | 2^(2+1)+2^2+2           |      5
 8 | 2^2^2                   |      3
 9 | 2^2^2+2                 |      4
10 | 2^2^2+2^2               |      5
11 | 2^2^2+2^2+2             |      6
12 | 2^2^2+2^(2+1)           |      5
13 | 2^2^2+2^(2+1)+2         |      6
14 | 2^2^2+2^(2+1)+2^2       |      7
15 | 2^2^2+2^(2+1)+2^2+2     |      8
16 | 2^(2^2+1)               |      3
17 | 2^(2^2+1)+2             |      4
18 | 2^(2^2+1)+2^2           |      5
19 | 2^(2^2+1)+2^2+2         |      6
20 | 2^(2^2+1)+2^(2+1)       |      5
21 | 2^(2^2+1)+2^(2+1)+2     |      6
22 | 2^(2^2+1)+2^(2+1)+2^2   |      7
23 | 2^(2^2+1)+2^(2+1)+2^2+2 |      8
24 | 2^(2^2+1)+2^2^2         |      6
25 | 2^(2^2+1)+2^2^2+2       |      7
26 | 2^(2^2+1)+2^2^2+2^2     |      8
27 | 2^(2^2+1)+2^2^2+2^2+2   |      9
28 | 2^(2^2+1)+2^2^2+2^(2+1) |      8
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A005245, A025280.

Programs

PARI

a(n)=if(n==0, 0, sum(k=0, logint(n,2), if(bittest(n,k), 1 + a((k+1)\2)))) \\ Andrew Howroyd, Apr 07 2023