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

Author

Chai Wah Wu, Dec 19 2024

Keywords

• nonn
• base

Links

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

Crossrefs

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

Programs

Mathematica

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

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])-(nA333516(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<
				

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