A100921 n appears A023416(n) times (appearances equal number of 0-bits).

0, 2, 4, 4, 5, 6, 8, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 32, 32, 32, 32, 32, 33, 33, 33, 33, 34, 34, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 37, 37

Offset: 0

Author

Rick L. Shepherd, Nov 21 2004

Keywords

• base
• easy
• nonn

Examples

			The binary representation of 16 is 10000, which has four 0-bits (and one 1-bit), hence 16 appears four times in this sequence (but only once in A100922).
		

Crossrefs

Cf. A100922 (n's appearances equal its number of 1-bits), A030530 (n's appearances equal its total number of bits), A023416, A059009.

Programs

Mathematica

Flatten[Table[Table[n, {DigitCount[n, 2, 0]}], {n, 0, 37}]] (* Amiram Eldar, Feb 18 2024 *)

Python

def A059015(n): return 2+(n+1)*((t:=(n+1).bit_length())-n.bit_count())-(1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)
def A100921(n):
    if n == 0: return 0
    m, k = 1, 1
    while A059015(m)<=n: m<<=1
    while m-k>1:
        r = m+k>>1
        if A059015(r)>n:
            m = r
        else:
            k = r
    return m  # Chai Wah Wu, Nov 11 2024

Formula

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/A059009(n) = 0.395592509... . - Amiram Eldar, Feb 18 2024