A004742 Numbers whose binary expansion does not contain 101.

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 19, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 48, 49, 50, 51, 56, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 76, 78, 79, 96, 97, 98, 99, 100, 102, 103, 112, 113, 114, 115, 120, 121, 124, 126, 127

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004743 (no 110), A003726 (no 111).

Programs

Haskell

a004742 n = a004742_list !! (n-1)
a004742_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 5 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Mathematica

Select[Range[0, 130], !StringContainsQ[IntegerString[#, 2], "101"] &] (* Amiram Eldar, Feb 13 2022 *)

PARI

is(n)=n=binary(n);for(i=3,#n,if(n[i]&&n[i-2]&&!n[i-1],return(0))); 1 \\ Charles R Greathouse IV, Mar 26 2013

PARI

is(n)=while(n>4, if(bitand(n,7)==5, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

PARI

is(n)=!bitand(bitand(n,n>>2),bitneg(n>>1)) \\ Charles R Greathouse IV, Oct 28 2021

PARI

searchLE(S,x)=my(t=setsearch(S,x)); if(t,t,setsearch(S,x,1)-1); \\ finds last element <= x
expand(~v, lim)=my(b=exponent(v[#v]+1), B=1<lim, listpop(~v));
list(lim)=lim\=1; if(lim<5, return(if(lim<0,[],[0..lim]))); my(v=List([0..3])); for(b=3,exponent(lim+1), expand(~v, 2^b-1)); expand(~v, lim); Vec(v)

Formula

Sum_{n>=2} 1/a(n) = 6.198475910942069028389983717965787117743378665090593775808705963863146498248... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022