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A100921
n appears A023416(n) times (appearances equal number of 0-bits).
1
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
0,2
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
KEYWORD
base,easy,nonn,changed
AUTHOR
Rick L. Shepherd, Nov 21 2004
STATUS
approved