A234957 Highest power of 4 dividing n.

1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16

Offset

1,4

Comments

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 4.

In the binary representation of n, remove zeros from the right until the number of zeros is even, then remove all but the rightmost one bit. - Ralf Stephan, Jan 05 2014

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Index entries for sequences related to binary expansion of n.

Formula

a(n) = 4^(valuation(n,4)).

a(n) = 4^(floor(valuation(n,2)/2)) = 4^A004526(A007814(n)). Recurrence: a(4n) = 4a(n), a(4n+k) = 1 for k=1,2,3. - Ralf Stephan, Jan 05 2014

G.f.: x/(1 - x) + 3 * Sum_{k>=1} 4^(k-1)*x^(4^k)/(1 - x^(4^k)). - Ilya Gutkovskiy, Jul 10 2019

From Amiram Eldar, Dec 31 2022: (Start)

Multiplicative with a(2^e) = 2^(2*floor(e/2)), and a(p^e) = 1 if p >= 3.

Dirichlet g.f.: zeta(s)*(4^s-1)/(4^s-4).

Sum_{k=1..n} a(k) ~ (3/(8*log(2)))*n*log(n) + (5/8 + 3*(gamma-1)/(8*log(2)))*n, where gamma is Euler's constant (A001620). (End)

Example

Since 8=4*2, then a(8)=4. Likewise, since 4 does not divide 9, a(9)=1.

Mathematica

Table[4^(IntegerExponent[n, 4]), {n, 1, 50}] (* G. C. Greubel, Apr 13 2017 *)

Prog

(Sage)
n=200 #change n for more terms
[4^(valuation(i, 4)) for i in [1..n]]
(PARI) a(n)=4^valuation(n, 4) \\ Charles R Greathouse IV, Aug 05 2015
(Python)
def A234957(n): return 1<<((~n&n-1).bit_length()&-2) # Chai Wah Wu, Jul 08 2022

Crossrefs

Cf. A001620, A006519, A038500.

Sequence in context: A366523 A365635 A276330 * A327939 A365837 A369669

Adjacent sequences: A234954 A234955 A234956 * A234958 A234959 A234960

Keyword

nonn,easy,mult

Author

Tom Edgar, Jan 01 2014

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

Status

approved