cp's OEIS Frontend

A359194 Binary complement of 3*n.

%I A359194 #48 Mar 24 2025 05:54:48
%S A359194 1,0,1,6,3,0,13,10,7,4,1,30,27,24,21,18,15,12,9,6,3,0,61,58,55,52,49,
%T A359194 46,43,40,37,34,31,28,25,22,19,16,13,10,7,4,1,126,123,120,117,114,111,
%U A359194 108,105,102,99,96,93,90,87,84,81,78,75,72,69,66,63,60,57
%N A359194 Binary complement of 3*n.
%C A359194 The binary complement takes the binary value of a number and turns any 1s to 0s and vice versa. This is equivalent to subtracting from the next larger Mersenne number.
%C A359194 It is currently unknown whether every starting positive integer, upon iteration, reaches 0.
%C A359194 From _M. F. Hasler_, Dec 26 2022: (Start)
%C A359194 This map enjoys the following properties:
%C A359194 (P1) a(2*n) = a(n)*2 + 1 (since 3*(2*n) is 3*n shifted one binary digit to the left, and the one's complement yields that of 3*n with a '1' appended).
%C A359194 (P2) As an immediate consequence of (P1), all even-indexed values are odd.
%C A359194 (P3) Also from (P1), by immediate induction we have a(2^n) = 2^n-1 for all n >= 0.
%C A359194 (P4) Also from (P1), a(4*n) = a(n)*4 + 3.
%C A359194 (P5) Similarly, a(4*n+1) = a(n)*4 (because the 1's complement of 3 is 0).
%C A359194 (P6) From (P5), a(n) = 0 for all n in A002450 (= (4^k-1)/3). [For the initial value at n = 0 the discrepancy is explained by the fact that the number 0 should be considered to have zero digits, but here the result is computed with 0 considered to have one binary digit.] (End)
%H A359194 Michael S. Branicky, <a href="/A359194/b359194.txt">Table of n, a(n) for n = 0..10000</a>
%H A359194 Joshua Searle, <a href="https://qedscience.wordpress.com/2021/02/19/collatz-inspired-sequences/">Collatz-inspired sequences</a>.
%F A359194 a(n) = A035327(3*n).
%F A359194 a(n) = 0 iff n belongs to A002450 \ {0}. - _Rémy Sigrist_, Dec 22 2022
%e A359194 a(7) = 10 because 3*7 = 21 = 10101_2, whose binary complement is 01010_2 = 10.
%e A359194 a(42) = 1 because 3*42 = 126 = 1111110_2, whose binary complement is 0000001_2 = 1.
%e A359194 a(52) = 99 by
%e A359194   3*n        = binary 10011100
%e A359194   complement = binary 01100011 = 99.
%o A359194 (Python)
%o A359194 def a(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length()) - 1)
%o A359194 print([a(n) for n in range(67)]) # _Michael S. Branicky_, Dec 20 2022
%o A359194 (PARI) a(n)=if(n, bitneg(3*n, exponent(3*n)+1), 1) \\ _Rémy Sigrist_, Dec 22 2022
%Y A359194 Trisection of A035327.
%Y A359194 Cf. A002450, A020988 (indices of 1's).
%Y A359194 Cf. A256078.
%K A359194 nonn,base,easy
%O A359194 0,4
%A A359194 _Joshua Searle_, Dec 19 2022