A359194 Binary complement of 3n.

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, 46, 43, 40, 37, 34, 31, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, 126, 123, 120, 117, 114, 111, 108, 105, 102, 99, 96, 93, 90, 87, 84, 81, 78, 75, 72, 69, 66, 63, 60, 57

Offset

0,4

Comments

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.

It is currently unknown whether every starting positive integer, upon iteration, reaches 0.

From M. F. Hasler, Dec 26 2022: (Start)

This map enjoys the following properties:

(P1) a(2n) = a(n)*2 + 1 (since 3*(2n) is 3*n shifted one binary digit to the left, and the one's complement yields that of 3*n with a '1' appended.

(P2) As an immediate consequence of (P1), all even-indexed values are odd.

(P3) Also from (P1), by immediate induction we have a(2^n) = 2^n-1 for all n >= 0.

(P4) Also from (P1), a(4n) = a(n)*4 + 3.

(P5) Similarly, a(4n+1) = a(n)*4 (because the 1's complement of 3 is 0)/

(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)

Links

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Joshua Searle, Collatz-inspired sequences.

Formula

a(n) = A035327(3n).

a(n) = 0 iff n belongs to A002450 \ {0}. - Rémy Sigrist, Dec 22 2022

Example

a(7) = 10 because 3*7 = 21 = 10101_2, whose binary complement is 01010_2 = 10.
a(42) = 1 because 3*42 = 126 = 1111110_2, whose binary complement is 0000001_2 = 1.
a(52) = 99 by
  3*n        = binary 10011100
  complement = binary 01100011 = 99.

Prog

(Python)
def a(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length()) - 1)
print([a(n) for n in range(67)]) # Michael S. Branicky, Dec 20 2022
(PARI) a(n)=if(n, bitneg(3*n, exponent(3*n)+1), 1) \\ Rémy Sigrist, Dec 22 2022

Crossrefs

Trisection of A035327.

Cf. A002450, A020988 (indices of 1's).

Cf. A256078.

Sequence in context: A153459 A102525 A119923 * A204420 A331570 A102410

Adjacent sequences: A359191 A359192 A359193 * A359195 A359196 A359197

Keyword

nonn,base,easy

Author

Joshua Searle, Dec 19 2022

Status

approved