A287896 a(n) = A002487(n)*A001511(n).

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

Offset: 1

Author

I. V. Serov, Jun 02 2017

Keywords

• nonn

Comments

Proposed name: N-fusc.

Each number n>0 appears in this sequence exactly n times.

From Yosu Yurramendi, Apr 08 2019: (Start)

The terms (n>0) may be written as a left-justified array with rows of length 2^m:

1,

2, 2,

3, 3, 4, 3,

4, 4, 6, 5, 6, 5, 6, 4,

5, 5, 8, 7, 9, 8, 10, 7, 8, 7, 10, 8, 9, 7, 8, 5,

6, 6, 10, 9, 12, 11, 14, 10, 12, 11, 16, 13, 15, 12, 14, 9, 10, 9, ...

...

as well as right-justified fashion:

1,

2, 2,

3, 3, 4, 3,

4, 4, 6, 5, 6, 5, 6, 4,

5, 5, 8, 7, 9, 8, 10, 7, 8, 7, 10, 8, 9, 7, 8, 5,

... 14, 9, 10, 9, 14, 12, 15, 13, 16, 11, 12, 10, 14, 11, 12, 9, 10, 6,

From these two dispositions interesting properties can be induced (see FORMULA section)

(End)

Links

• I. V. Serov, Table of n, a(n) for n = 1..8192

Crossrefs

Cf. A001511, A002487, A007306, A288002.

Programs

Mathematica

Table[Block[{a = 1, b = 0, m = n}, While[m > 0, If[OddQ@ m, b = a + b, a = a + b]; m = Floor[m/2]]; b] IntegerExponent[2 n, 2], {n, 89}] (* Michael De Vlieger, Jun 14 2017, after Jean-François Alcover at A002487 *)

Python

from functools import reduce
def A287896(n): return (n&-n).bit_length()*sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, Jul 14 2022

Formula

a(1) = 1; for n>1: a(n) = (A002487(n-1) + A002487(n) + A002487(n+1))/2.

a(n) = A007306(n) - A288002(n).

From Yosu Yurramendi, Apr 08 2019: (Start)

For m >= 0, 0 <= k < 2^m, a(2^(m+1)+k) - a(2^m+k) = a(k). a(0) = 1 is needed.

For m >= 0, 0 <= k < 2^m, a(2^(m+1)-1-k) - a(2^(m)-1-k) = a(k).

(End)