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

1,2

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

From Jules Beauchamp, Sep 10 2025: (Start)

This sequence includes all the squares. It also includes the sequence A016813 (numbers of the form 4*k + 1). Composite terms that are not multiples of previous terms are multiples of terms not already in the sequence.

All odd terms > 1 are sums of an odd and even already in the sequence, and apart from those of the form 4^m, all even terms are the sum of two previous even terms (but not the sum of two odd terms).

This sequence is fractal. For any term, k, divide all multiples of k in the sequence by k to return the original sequence.

If odd k is in the sequence then so also is 2*k - 1. If even k is in the sequence then so also is 2*k + 1. If k is in the sequence then so also is 4^m*k, 4^m*k + 1 and 4^m*k - 3. (End)

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Index entries for 2-automatic sequences.

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 *)

Prog

(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

Crossrefs

Cf. A065882, A036554, A055050.

Sequence in context: A034705 A006844 A022425 * A331531 A363319 A363267

Adjacent sequences: A277546 A277547 A277548 * A277550 A277551 A277552

Keyword

nonn,easy

Author

Clark Kimberling, Oct 20 2016

Status

approved