A379269 Numbers whose binary representation has exactly three zeros.

8, 17, 18, 20, 24, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217

Offset

1,1

Links

Table of n, a(n) for n=1..58.

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

Formula

a(n) = (A360573(n)-1)/2.

A023416(a(n)) = 3.

Let a = floor((24n)^(1/4))+3 if n>binomial(floor((24n)^(1/4))+2,4) and a = floor((24n)^(1/4))+2 otherwise. Let j = binomial(a,4)-n. Then a(n) = 2^a-1-2^(A360010(j+1)+1)-2^(A056557(j)+1)-2^(A333516(j+1)-1).

Sum_{n>=1} 1/a(n) = 1.3949930090659130972172214185888677947877214389482588641632435250211546702139813215203065255971026537... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Dec 21 2024

Mathematica

Select[Range[2^8], Count[IntegerDigits[#, 2], 0]==3&] (* James C. McMahon, Dec 20 2024 *)

Prog

(Python)
from math import comb, isqrt
from sympy import integer_nthroot
def A056557(n): return (k:=isqrt(r:=n+1-comb((m:=integer_nthroot(6*(n+1), 3)[0])-(n<comb(m+2, 3))+2, 3)<<1))-((r<<2)<=(k<<2)*(k+1)+1)
def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))+1, 3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)), 2)+1
def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
def A379269(n):
    a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+2
    j = comb(a, 4)-n
    b, c, d = A360010(j+1)+1, A056557(j)+1, A333516(j+1)-1
    return (1<<a)-(1<<b)-(1<<c)-(1<<d)-1

Crossrefs

Cf. A023416, A056557, A333516, A360010, A360573.

Sequence in context: A066554 A302976 A244537 * A046459 A274770 A075485

Adjacent sequences: A379266 A379267 A379268 * A379270 A379273 A379274

Keyword

nonn,base,new

Author

Chai Wah Wu, Dec 19 2024

Status

approved