A277549 Numbers k such that k/4^m == 1 (mod 4), where 4^m is the greatest power of 4 that divides k.

1, 4, 5, 9, 13, 16, 17, 20, 21, 25, 29, 33, 36, 37, 41, 45, 49, 52, 53, 57, 61, 64, 65, 68, 69, 73, 77, 80, 81, 84, 85, 89, 93, 97, 100, 101, 105, 109, 113, 116, 117, 121, 125, 129, 132, 133, 137, 141, 144, 145, 148, 149, 153, 157, 161, 164, 165, 169, 173

Offset: 1

Author

Clark Kimberling, Oct 20 2016

Keywords

• nonn
• easy

Comments

Positions of 1 in A065882.

This is one sequence in a 3-way splitting of the positive integers; the other two are A036554 and A055050, as in the Mathematica program.

The asymptotic density of this sequence is 1/3. - Amiram Eldar, Mar 08 2021

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A065882, A036554, A055050.

Programs

Maple

filter:= n -> n/2^(2*floor(padic:-ordp(n,2)/2)) mod 4 = 1:
select(filter, [$1..1000]); # Robert Israel, Oct 20 2016

Mathematica

z = 160; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}];
p[b_, d_] := Flatten[Position[a[b], d]];
p[4, 1] (* A277549 *)
p[4, 2] (* A036554 *)
p[4, 3] (* A055050 *)

PARI

isok(n) = n/4^valuation(n,4) % 4 == 1; \\ Michel Marcus, Oct 20 2016

Python

from itertools import count, islice
def A277549_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n>>((~n&n-1).bit_length()&-2))&3==1,count(max(startvalue,1)))
A277549_list = list(islice(A277549_gen(),30)) # Chai Wah Wu, Jul 09 2022

Python

def A277549(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-1>>2)+1 for i in range((x.bit_length()>>1)+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 19 2025