A191257 - OEIS

%I #67 Feb 17 2025 14:40:45

%S 1,3,5,7,8,9,11,13,15,17,19,21,23,24,25,27,29,31,33,35,37,39,40,41,43,

%T 45,47,49,51,53,55,56,57,59,61,63,64,65,67,69,71,72,73,75,77,79,81,83,

%U 85,87,88,89,91,93,95,97,99,101,103,104,105,107,109,111,113,115,117,119,120,121,123,125,127,129,131,133,135,136,137,139,141,143

%N a(n) = A067368(n)/2.

%C From _Jianing Song_, Sep 21 2018: (Start)

%C Numbers n such that A191255(n) = 0 or 3. Previous definition was numbers n such that A191255(2*n) = 1, that is, numbers of the form 2^(3t)*s where s is an odd number.

%C {+-a(n)} gives all nonzero cubes modulo all powers of 2, that is, nonzero cubes over the 2-adic integers. So this sequence is closed under multiplication. (End)

%C The old entry had the conjecture that a(n) = A067368(n)/2. _Jianing Song_, Sep 21 2018 showed that this is true, and gave us the simpler definition that we have now used. The conjecture is correct because {a(n)} lists the numbers of the form 2^(3t)*s, and {A067368(n)} lists the numbers of the form 2^(3t+1)*s, where s is an odd number. Note also that a(n) = A213258(n)/4.

%C The asymptotic density of this sequence is 4/7. - _Amiram Eldar_, May 31 2024

%H Recto Rex M. Calingasan and Alexander Vincent B. Policarpio, <a href="https://doi.org/10.1063/1.5012157">On the zeros of the OEIS A191257 zeta function</a>, AIP Conference Proceedings 1905, 030011 (2017).

%t t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3},

%t       3 -> {0, 1}}] &, {0}, 9] (* A191255 *)

%t Flatten[Position[t, 0]] (* A005408, the odds *)

%t a = Flatten[Position[t, 1]] (* A067368 *)

%t b = Flatten[Position[t, 2]] (* A213258 *)

%t a/2  (* A191257 *)

%t b/4  (* a/2 *)

%o (PARI) isok(n) = valuation(2*n, 2)%3==1; \\ _Altug Alkan_, Sep 21 2018

%o (Python)

%o def A191257(n):

%o     def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))

%o     m, k = n, f(n)

%o     while m != k: m, k = k, f(k)

%o     return m # _Chai Wah Wu_, Feb 17 2025

%Y Cf. A067368, A191255, A213258.

%Y Perfect powers over the 2-adic integers:

%Y Squares: positive: A234000; negative: A004215 (negated);

%Y Cubes: this sequence;

%Y Fourth powers: positive: A319281; negative: A319282 (negated).

%K nonn

%O 1,2

%A _Clark Kimberling_, May 28 2011

%E Name corrected by _Altug Alkan_, Apr 03 2018

%E New name from _Jianing Song_, Sep 21 2018