A100921 - OEIS

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

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..72.

FORMULA

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

EXAMPLE

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).

MATHEMATICA

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

PROG

(Python)

def A059015(n): return 2+(n+1)*((t:=(n+1).bit_length())-n.bit_count())-(1<<t)-(sum((m:=1<<j)*((k:=n>>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

CROSSREFS

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

Sequence in context: A071192 A295012 A308629 * A140201 A351393 A057861

Adjacent sequences: A100918 A100919 A100920 * A100922 A100923 A100924

KEYWORD

base,easy,nonn

AUTHOR

Rick L. Shepherd, Nov 21 2004

STATUS

approved