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A379269 Numbers whose binary representation has exactly three zeros.

%I A379269 #15 Dec 21 2024 11:09:29
%S A379269 8,17,18,20,24,35,37,38,41,42,44,49,50,52,56,71,75,77,78,83,85,86,89,
%T A379269 90,92,99,101,102,105,106,108,113,114,116,120,143,151,155,157,158,167,
%U A379269 171,173,174,179,181,182,185,186,188,199,203,205,206,211,213,214,217
%N A379269 Numbers whose binary representation has exactly three zeros.
%H A379269 Robert Baillie and Thomas Schmelzer, <a href="https://library.wolfram.com/infocenter/MathSource/7166/">Summing Kempner's Curious (Slowly-Convergent) Series</a>, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
%F A379269 a(n) = (A360573(n)-1)/2.
%F A379269 A023416(a(n)) = 3.
%F A379269 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).
%F A379269 Sum_{n>=1} 1/a(n) = 1.3949930090659130972172214185888677947877214389482588641632435250211546702139813215203065255971026537... (calculated using Baillie's irwinSums.m, see Links). - _Amiram Eldar_, Dec 21 2024
%t A379269 Select[Range[2^8],Count[IntegerDigits[#,2],0]==3&] (* _James C. McMahon_, Dec 20 2024 *)
%o A379269 (Python)
%o A379269 from math import comb, isqrt
%o A379269 from sympy import integer_nthroot
%o A379269 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)
%o A379269 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
%o A379269 def A360010(n): return (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2, 3))
%o A379269 def A379269(n):
%o A379269     a = (a2:=integer_nthroot(24*n, 4)[0])+(n>comb(a2+2, 4))+2
%o A379269     j = comb(a,4)-n
%o A379269     b, c, d = A360010(j+1)+1, A056557(j)+1, A333516(j+1)-1
%o A379269     return (1<<a)-(1<<b)-(1<<c)-(1<<d)-1
%Y A379269 Cf. A023416, A056557, A333516, A360010, A360573.
%K A379269 nonn,base
%O A379269 1,1
%A A379269 _Chai Wah Wu_, Dec 19 2024