cp's OEIS Frontend

A191257 a(n) = A067368(n)/2.

%I A191257 #67 Feb 17 2025 14:40:45
%S A191257 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 A191257 45,47,49,51,53,55,56,57,59,61,63,64,65,67,69,71,72,73,75,77,79,81,83,
%U A191257 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 A191257 a(n) = A067368(n)/2.
%C A191257 From _Jianing Song_, Sep 21 2018: (Start)
%C A191257 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 A191257 {+-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 A191257 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 A191257 The asymptotic density of this sequence is 4/7. - _Amiram Eldar_, May 31 2024
%H A191257 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 A191257 t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3},
%t A191257       3 -> {0, 1}}] &, {0}, 9] (* A191255 *)
%t A191257 Flatten[Position[t, 0]] (* A005408, the odds *)
%t A191257 a = Flatten[Position[t, 1]] (* A067368 *)
%t A191257 b = Flatten[Position[t, 2]] (* A213258 *)
%t A191257 a/2  (* A191257 *)
%t A191257 b/4  (* a/2 *)
%o A191257 (PARI) isok(n) = valuation(2*n, 2)%3==1; \\ _Altug Alkan_, Sep 21 2018
%o A191257 (Python)
%o A191257 def A191257(n):
%o A191257     def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))
%o A191257     m, k = n, f(n)
%o A191257     while m != k: m, k = k, f(k)
%o A191257     return m # _Chai Wah Wu_, Feb 17 2025
%Y A191257 Cf. A067368, A191255, A213258.
%Y A191257 Perfect powers over the 2-adic integers:
%Y A191257 Squares: positive: A234000; negative: A004215 (negated);
%Y A191257 Cubes: this sequence;
%Y A191257 Fourth powers: positive: A319281; negative: A319282 (negated).
%K A191257 nonn
%O A191257 1,2
%A A191257 _Clark Kimberling_, May 28 2011
%E A191257 Name corrected by _Altug Alkan_, Apr 03 2018
%E A191257 New name from _Jianing Song_, Sep 21 2018