A038189 Bit to left of least significant 1-bit in binary expansion of n.
0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1
Offset: 0
Examples
a(6) = 1 since 6 = 110 and bit before rightmost 1 is a 1.
References
- Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003, sect. 5.1.6
Links
Crossrefs
Programs
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C
int a(int n) { return (n & ((n&-n)<<1)) ? 1 : 0; } /* from Russ Cox */ -
Magma
function a (n) if n eq 0 then return 0; // alternatively, return 1; else while IsEven(n) do n := n div 2; end while; end if; return n div 2 mod 2; end function; nlo := 0; nhi := 32; [a(n) : n in [nlo..nhi] ]; // Fred Lunnon, Mar 27 2018
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Maple
A038189 := proc(n) option remember; if n = 0 then 0 ; elif type(n,'even') then procname(n/2) ; elif modp(n,4) = 1 then 0 ; else 1 ; end if; end proc: seq(A038189(n),n=0..100) ; # R. J. Mathar, Mar 30 2018
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Mathematica
f[n_] := Block[{id2 = Join[{0}, IntegerDigits[n, 2]]}, While[ id2[[-1]] == 0, id2 = Most@ id2]; id2[[-2]]]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Apr 14 2009 and fixed Feb 27 2014 *) f[n_] := f[n] = Switch[Mod[n, 4], 0, f[n/2], 1, 0, 2, f[n/2], 3, 1]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Apr 14 2009, fixed Feb 27 2014 *) a[ n_] := If[n==0, 0, Mod[(n/2^IntegerExponent[n, 2]-1)/2, 2]]; (* Michael Somos, Oct 05 2024 *) -
PARI
{a(n) = if(n==0, 0, ((n/2^valuation(n,2)-1)/2)%2)}; /* Michael Somos, Sep 22 2005 */ -
PARI
a(n) = if(n<3, 0, prod(m=1,n, kronecker(-n,m)==kronecker(m,n))) /* Michael Somos, Sep 22 2005 */
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PARI
A038189(n)=bittest(n,valuation(n,2)+1) \\ M. F. Hasler, Aug 06 2015
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PARI
a(n)=my(h=bitand(n,-n));n=bitand(n,h<<1);n!=0; \\ Joerg Arndt, Apr 09 2021
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Python
def A038189(n): s = bin(n)[2:] m = len(s) i = s[::-1].find('1') return int(s[m-i-2]) if m-i-2 >= 0 else 0 # Chai Wah Wu, Apr 08 2021
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Python
def a(n): return 1 if (n & ((n&-n)<<1)) else 0 print([a(n) for n in range(108)]) # Michael S. Branicky, Feb 06 2025 after Russ Cox
Formula
a(0) = 0, a(2*n) = a(n) for n>0, a(4*n+1) = 0, a(4*n+3) = 1.
G.f.: Sum_{k>=0} t^3/(1-t^4), where t=x^2^k. Parity of A025480. For n >= 1, a(n) = 1/2 * (1 - (-1)^A025480(n-1)). - Ralf Stephan, Jan 04 2004 [index adjusted by Peter Munn, Jun 22 2022]
a(n) = 1 if Kronecker(-n,m)=Kronecker(m,n) for all m, otherwise a(n)=0. - Michael Somos, Sep 22 2005
a(n) = 1 iff A164677(n) < 0. - M. F. Hasler, Aug 06 2015
For n >= 1, a(n) = A065339(n) mod 2. - Peter Munn, Jun 20 2022
From A.H.M. Smeets, Mar 08 2023: (Start)
a(n+1) = 1 - A014577(n) for n >= 0.
a(n+1) = 2 - A014710(n) for n >= 0.
a(n) = (1 - A034947(n))/2 for n > 0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Aug 30 2024
Extensions
More terms from David W. Wilson
Definition corrected by Russ Cox and Ralf Stephan, Nov 08 2004
Comments