A030130 Binary expansion contains a single 0.

0, 2, 5, 6, 11, 13, 14, 23, 27, 29, 30, 47, 55, 59, 61, 62, 95, 111, 119, 123, 125, 126, 191, 223, 239, 247, 251, 253, 254, 383, 447, 479, 495, 503, 507, 509, 510, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1535, 1791, 1919, 1983, 2015, 2031, 2039

Offset: 1

Author

Toby Donaldson (tjdonald(AT)uwaterloo.ca)

Keywords

• nonn
• base
• easy
• look

Comments

From Reinhard Zumkeller, Aug 29 2009: (Start)

A023416(a(n)) = 1;

apart from the initial term the sequence can be seen as a triangle read by rows, see A164874;

A055010 and A086224 are subsequences, see also A000918 and A036563. (End)

Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - Zak Seidov, Aug 06 2010

Examples

			23 is OK because it is '10111' in base 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000918, A023416, A036563, A055010, A086224, A140977, A160502, A164874, A190620.

Programs

C

long int element (long int i) { return (pow(2,g(i))-1-pow(2,(pow(2*g(i)-1,2)-1-8*i)/8));} long int g(long int m) {if (m==0) return(1); return ((sqrt(8*m-7)+3)/2);}

Haskell

a030130 n = a030130_list !! (n-1)
a030130_list = filter ((== 1) . a023416) [0..]
-- Reinhard Zumkeller, Mar 31 2015, Dec 07 2012

Magma

[0] cat [k:k in [0..2050]| Multiplicity(Intseq(k,2),0) eq 1]; // Marius A. Burtea, Feb 06 2020

Mathematica

Sort[Flatten[{{0}, Table[2^n - 2^m - 1, {n, 2, 50}, {m, 0, n - 2}]}]] (* Zak Seidov, Aug 06 2010 *)
Select[Range[0,2100],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Dec 19 2021 *)

PARI

print1("0, ");for(k=1,2039,my(v=digits(k,2));if(vecsum(v)==#v-1,print1(k,", "))) \\ Hugo Pfoertner, Feb 06 2020

Python

from math import isqrt, comb
def A030130(n): return (1<<(a:=(isqrt(n-1<<3)+1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

Formula

a(n) = 2^(g(n))-1-2^(((2*g(n)-1)^2-1-8*n)/8) with g(n)=int((sqrt(8*n-7)+3)/2) for all n>0 and g(0)=1. - Ulrich Schimke (ulrschimke(AT)aol.com)

a(n+1) = A140977(a(n)) for any n > 1. - Rémy Sigrist, Feb 06 2020

Sum_{n>=2} 1/a(n) = A160502. - Amiram Eldar, Oct 06 2020

a(n) = (A190620(n-1)-1)/2. - Chai Wah Wu, Dec 19 2024

Extensions

More terms from Erich Friedman

Offset fixed by Reinhard Zumkeller, Aug 24 2009