A030130 - OEIS

%I #35 Sep 08 2022 08:44:50

%S 0,2,5,6,11,13,14,23,27,29,30,47,55,59,61,62,95,111,119,123,125,126,

%T 191,223,239,247,251,253,254,383,447,479,495,503,507,509,510,767,895,

%U 959,991,1007,1015,1019,1021,1022,1535,1791,1919,1983,2015,2031,2039

%N Binary expansion contains a single 0.

%C From _Reinhard Zumkeller_, Aug 29 2009: (Start)

%C A023416(a(n)) = 1;

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

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

%C Zero and numbers of form 2^m-2^k-1, 2 <= m, 0 <= k <= m-2. - _Zak Seidov_, Aug 06 2010

%H Reinhard Zumkeller, <a href="/A030130/b030130.txt">Table of n, a(n) for n = 1..1000</a>

%F 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)

%F a(n+1) = A140977(a(n)) for any n > 1. - _Rémy Sigrist_, Feb 06 2020

%F Sum_{n>=2} 1/a(n) = A160502. - _Amiram Eldar_, Oct 06 2020

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

%t Sort[Flatten[{{0}, Table[2^n - 2^m - 1, {n, 2, 50}, {m, 0, n - 2}]}]] (* _Zak Seidov_, Aug 06 2010 *)

%t Select[Range[0,2100],DigitCount[#,2,0]==1&] (* _Harvey P. Dale_, Dec 19 2021 *)

%o (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);}

%o (Haskell)

%o a030130 n = a030130_list !! (n-1)

%o a030130_list = filter ((== 1) . a023416) [0..]

%o -- _Reinhard Zumkeller_, Mar 31 2015, Dec 07 2012

%o (PARI) print1("0, ");for(k=1,2039,my(v=digits(k,2));if(vecsum(v)==#v-1,print1(k,", "))) \\ _Hugo Pfoertner_, Feb 06 2020

%o (Magma) [0] cat [k:k in [0..2050]| Multiplicity(Intseq(k,2),0) eq 1]; // _Marius A. Burtea_, Feb 06 2020

%Y Cf. A000918, A023416, A036563, A055010, A086224, A140977, A160502, A164874.

%K nonn,base,easy,look

%O 1,2

%A Toby Donaldson (tjdonald(AT)uwaterloo.ca)

%E More terms from _Erich Friedman_

%E Offset fixed by _Reinhard Zumkeller_, Aug 24 2009