A213258 Positive integers that are not in A213257.

4, 12, 20, 28, 32, 36, 44, 52, 60, 68, 76, 84, 92, 96, 100, 108, 116, 124, 132, 140, 148, 156, 160, 164, 172, 180, 188, 196, 204, 212, 220, 224, 228, 236, 244, 252, 256, 260, 268, 276, 284, 288, 292, 300, 308, 316, 324, 332, 340, 348, 352, 356, 364, 372, 380, 388, 396, 404, 412, 416, 420, 428, 436, 444, 452, 460, 468, 476, 480, 484, 492, 500

Offset: 1

Author

John W. Layman, Jun 07 2012

Keywords

• nonn

Comments

Conjecture. The terms of this sequence are given by the positions of 2 in the fixed-point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01 (see A191255). (This has been confirmed for over 5000 terms of A213257.) To illustrate, the fixed-point of the indicated morphism is {0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,1,0,1,0,2,0,...} and 2 occurs at positions {4,12,20,...}, integers in this sequence but missing from A213257.

It appears that the terms of this sequence are all of the form of 4 times an odd integer multiplied by a nonnegative power of 8.

The above two conjectures are correct. This is indeed positions of 2 in A191255, and numbers of the form 2^(3t+2)*s where s is an odd number. - Jianing Song, Sep 21 2018

The asymptotic density of this sequence is 1/7. - Amiram Eldar, May 31 2024

Crossrefs

Cf. A067368, A191255, A191257, A213257.

Programs

Mathematica

Select[Range[500], Mod[IntegerExponent[#, 2], 3] == 2 &] (* Amiram Eldar, May 31 2024 *)

PARI

is(n) = valuation(n, 2) % 3 == 2; \\ Amiram Eldar, May 31 2024

Python

def A213258(n):
    def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<2 # Chai Wah Wu, Feb 17 2025

Formula

a(n) = 2*A067368(n) = 4*A191257(n). - Amiram Eldar, May 31 2024