cp's OEIS Frontend

A091067 Numbers whose odd part is of the form 4k+3.

%I A091067 #86 Aug 31 2024 04:31:50
%S A091067 3,6,7,11,12,14,15,19,22,23,24,27,28,30,31,35,38,39,43,44,46,47,48,51,
%T A091067 54,55,56,59,60,62,63,67,70,71,75,76,78,79,83,86,87,88,91,92,94,95,96,
%U A091067 99,102,103,107,108,110,111,112,115,118,119,120,123,124,126,127,131
%N A091067 Numbers whose odd part is of the form 4k+3.
%C A091067 Either of form 2*a(m) or 4k+3, k >= 0, 0 < m < n.
%C A091067 A000265(a(n)) is an element of A004767.
%C A091067 a(n) such that A038189(a(n)) = 1.
%C A091067 Numbers n such that Kronecker(-n, m) = Kronecker(m, n) for all m. - _Michael Somos_, Sep 22 2005
%C A091067 From _Antti Karttunen_, Feb 20-21 2015: (Start)
%C A091067 Gives all n for which A005811(n) - A005811(n-1) = -1, from which follows that a(n) = the least k such that A255070(k) = n.
%C A091067 Gives the positions of even terms in A003602. (End)
%C A091067 Indices of negative terms in A164677. - _M. F. Hasler_, Aug 06 2015
%C A091067 Indices of the 0's in A014577. - _Gabriele Fici_, Jun 02 2016
%C A091067 Also indices of -1 in A034947. - _Jianing Song_, Apr 24 2021
%C A091067 Conjecture: alternate definition of same sequence is that a(1)=3 and a(n) is the smallest number > a(n-1) so that no number that is the sum of at most 2 terms in this sequence is a power of 2. - _J. Lowell_, Jan 20 2024
%C A091067 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Aug 31 2024
%H A091067 Reinhard Zumkeller, <a href="/A091067/b091067.txt">Table of n, a(n) for n = 1..10000</a>
%H A091067 J.-P. Allouche and J. Shallit, <a href="https://arxiv.org/abs/2006.04708">On three conjectures of P. Barry</a>, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
%H A091067 Paul Barry, <a href="https://arxiv.org/abs/2005.04066">Some observations on the Rueppel sequence and associated Hankel determinants</a>, arXiv:2005.04066 [math.CO], 2020.
%H A091067 Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index TurnRight, with a(n) = TurnRight(n-1).
%F A091067 a(n) = A060833(n+1) - 1. [See _N. Sato_'s Feb 12 2013 comment in A060833.]
%F A091067 Other identities. For all n >= 1 it holds that:
%F A091067 A014707(a(n) + 1) = 1. - _Reinhard Zumkeller_, Sep 28 2011
%F A091067 A055975(a(n)) < 0. - _Reinhard Zumkeller_, Apr 28 2012
%F A091067 From _Antti Karttunen_, Feb 20-21 2015: (Start)
%F A091067 a(n) = A246590(n)/2.
%F A091067 A255070(a(n)) = n, or equally, A236840(a(n)) = 2n.
%F A091067 a(n) = 1 + A255068(n-1). (End)
%t A091067 Select[Range[150], Mod[# / 2^IntegerExponent[#, 2], 4] == 3 &] (* _Amiram Eldar_, Aug 31 2024 *)
%o A091067 (PARI) for(n=1,200,if(((n/2^valuation(n,2)-1)/2)%2,print1(n",")))
%o A091067 (PARI) {a(n) = local(m, c); if( n<1, 0, c=0; m=1; while( c<n, m++; if( ((m/2^valuation(m, 2) - 1) / 2) % 2, c++)); m)}; /* _Michael Somos_, Sep 22 2005 */
%o A091067 (PARI) is_A091067(n)=bittest(n,valuation(n,2)+1) \\ _M. F. Hasler_, Aug 06 2015
%o A091067 (PARI) a(n) = my(t=1); n<<=1; forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n++;t=!t)); n; \\ _Kevin Ryde_, Mar 21 2021
%o A091067 (Haskell)
%o A091067 import Data.List (elemIndices)
%o A091067 a091067 n = a091067_list !! (n-1)
%o A091067 a091067_list = map (+ 1) $ elemIndices 1 a014707_list
%o A091067 -- _Reinhard Zumkeller_, Sep 28 2011
%o A091067 (Scheme, with Antti Karttunen's IntSeq-library, two versions)
%o A091067 (define A091067 (MATCHING-POS 1 1 (COMPOSE even? A003602)))
%o A091067 (define A091067 (NONZERO-POS 1 0 A038189))
%o A091067 ;; _Antti Karttunen_, Feb 20 2015
%Y A091067 Essentially one less than A060833.
%Y A091067 Characteristic function: A038189.
%Y A091067 Complement of A091072.
%Y A091067 First differences are in A106836 (from its second term onward).
%Y A091067 Sequence A246590 gives the even terms.
%Y A091067 Gives the positions of records (after zero) for A255070 (equally, the position of the first n there).
%Y A091067 Cf. A106837 (gives n such that both n and n+1 are terms of this sequence).
%Y A091067 Cf. A098502 (gives n such that both n and n+2 are, but n+1 is not in this sequence).
%Y A091067 Cf. also A000265, A003602, A004767, A005811, A014707, A034947, A055975, A236840, A255068, A255327, A255330.
%K A091067 nonn,easy
%O A091067 1,1
%A A091067 _Ralf Stephan_, Feb 22 2004