OFFSET
1,4
COMMENTS
From Peter Luschny, May 31 2009: (Start)
Let odd(n) be the characteristic function of the odd numbers (A000035) and sigma(n) the number of 1's in binary expansion of n (A000120). Then a(n) = 2^(sigma(n-1) - sigma(n) + odd(n)).
Let B_{n} be the Bernoulli number. Then this sequence is also
a(n) = denominator(4*(4^n-1)*B_{2*n}/n). (End)
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = 2^max(A007814(n) - 1, 0). - Max Alekseyev, Feb 09 2011
a((2*n-1)*2^p) = A011782(p), p >= 0 and n >= 1. - Johannes W. Meijer, Jan 25 2013
a(n) = (1 + A140670(n))/2. - Antti Karttunen, Nov 18 2017
From Amiram Eldar, Dec 31 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s-2+1/2^s)/(2^s-2).
Sum_{k=1..n} a(k) ~ (1/(4*log(2)))*n*log(n) + (5/8 + (gamma-1)/(4*log(2)))*n, where gamma is Euler's constant (A001620). (End)
MAPLE
nmax:=96: p:= floor(log[2](nmax)): for n from 1 to nmax do a(n):=1 end do: for q from 1 to p do for n from 1 to nmax do if n mod 2^q = 0 then a(n):= 2^(q-1) end if: end do: end do: seq(a(n), n=1..nmax);
From Peter Luschny, May 31 2009: (Start)
a := proc(n) local sigma; sigma := proc(n) local i; add(i, i=convert(n, base, 2)) end; 2^(sigma(n-1)-sigma(n)+`if`(type(n, odd), 1, 0)) end: seq(a(n), n=1..96);
a := proc(n) denom(4*(4^n-1)*bernoulli(2*n)/n) end: seq(a(n), n=1..96); (End)
MATHEMATICA
a[n_] := If[OddQ[n], 1, 2^(IntegerExponent[n, 2] - 1)]; Array[a, 100] (* Amiram Eldar, Jul 02 2020 *)
PROG
(PARI) A160467(n) = 2^max(valuation(n, 2)-1, 0); \\ Antti Karttunen, Nov 18 2017, after Max Alekseyev's Feb 09 2011 formula.
(Python)
def A160467(n): return max(1, (n&-n)>>1) # Chai Wah Wu, Jul 08 2022
CROSSREFS
KEYWORD
base,easy,nonn,mult
AUTHOR
Johannes W. Meijer, May 24 2009, Jun 28 2011
EXTENSIONS
Keyword mult added by Max Alekseyev, Feb 09 2011
Name changed by Antti Karttunen, Nov 18 2017
STATUS
approved