A160467 a(n) = 1 if n is odd; otherwise, a(n) = 2^(k-1) where 2^k is the largest power of 2 that divides n.

1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 32, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16

Offset: 1

Author

Johannes W. Meijer, May 24 2009, Jun 28 2011

Keywords

• base
• easy
• nonn
• mult

Comments

Fifth factor of the row sums A160466 of the Eta triangle A160464.

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

Crossrefs

Cf. A000265, A001620, A007814, A026741, A140670, A160464, A220466, A006519, A000010.

Programs

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 *)

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

Formula

a(n) = A026741(n)/A000265(n). - Paul Curtz, Apr 18 2010

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)

a(n) = A006519(n)/gcd(n,2). - Ridouane Oudra, Feb 08 2025

a(n) = A000010(A006519(n)). - Ridouane Oudra, Jul 27 2025

Extensions

Keyword mult added by Max Alekseyev, Feb 09 2011

Name changed by Antti Karttunen, Nov 18 2017