A343500 Positions of 2's in A003324.

2, 8, 10, 12, 18, 26, 28, 32, 34, 40, 42, 44, 48, 50, 58, 60, 66, 72, 74, 76, 82, 90, 92, 98, 104, 106, 108, 112, 114, 122, 124, 128, 130, 136, 138, 140, 146, 154, 156, 160, 162, 168, 170, 172, 176, 178, 186, 188, 192, 194, 200, 202, 204, 210, 218, 220, 226

Offset: 1

Author

Jianing Song, Apr 17 2021

Keywords

• nonn
• easy

Comments

Numbers of the form (2*k+1) * 2^e where e >= 1, k+e is odd. In other words, union of {(4*m+1) * 2^(2t-1)} and {(4*m+3) * 2^(2t)}, where m >= 0, t > 0.

Numbers whose quaternary (base-4) expansion ends in 300...00 or 0200..00 or 2200..00. At least one trailing zero is required in the first case but not in the latter two cases.

There are precisely 2^(N-2) terms <= 2^N for every N >= 2.

Also even indices of -1 in A209615. - Jianing Song, Apr 24 2021

Complement of A343501 with respect to the even numbers. - Jianing Song, Apr 26 2021

Examples

			A003324 starts with 1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 4, ... We have A003324(2) = A003324(8) = A003324(10) = A003324(12) = ... = 2, so this sequence starts with 2, 8, 10, 12, ...
		

Links

• Jianing Song, Table of n, a(n) for n = 1..16384 (all terms <= 2^16).

Crossrefs

Cf. A003324, A343501 (positions of 4's), A209615, A338692.

Even terms in A338691.

Programs

Mathematica

okQ[n_] := If[OddQ[n], False, Module[{e = IntegerExponent[n, 2], k}, k = (n/2^e - 1)/2; OddQ[k + e]]];
Select[Range[1000], okQ] (* Jean-François Alcover, Apr 19 2021, after PARI *)

PARI

isA343500(n) = if(n%2, 0, my(e=valuation(n, 2), k=bittest(n,e+1)); (k+e)%2)

Python

def A343500(n):
    def f(x): return n+x-sum(((x>>i)-1>>2)+1 for i in range(1,x.bit_length(),2))-sum(((x>>i)-3>>2)+1 for i in range(2,x.bit_length(),2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 25 2025

Formula

a(n) = 2*A338692(n). - Hugo Pfoertner, Apr 26 2021