A380008 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are odious numbers (A000069).

0, 2, 4, 6, 16, 18, 20, 22, 128, 130, 132, 134, 144, 146, 148, 150, 256, 258, 260, 262, 272, 274, 276, 278, 384, 386, 388, 390, 400, 402, 404, 406, 2048, 2050, 2052, 2054, 2064, 2066, 2068, 2070, 2176, 2178, 2180, 2182, 2192, 2194, 2196, 2198, 2304, 2306, 2308, 2310, 2320, 2322, 2324, 2326, 2432, 2434, 2436, 2438, 2448, 2450, 2452, 2454

Offset

0,2

Comments

These t in binary representation have 1s only in positions with 0s in the Thue-Morse sequence (A010059) with beginning of that sequence corresponding to least significant bit. a(n) can be derived from n by placing the bits of n into a(n) at those permitted positions.

a(n) can be represented in base 4 equal to binary representation of n with each digit multiplied by 1 or 2 according to the 1-2 Thue-Morse sequence A001285 starting in the least significant digit and transforming 1->2, and 2->1.

Any pair 2*p and 2*p+1 has one evil and the other odious number, so the bit at position p in n goes to either 2*p or 2*p+1 in a(n), according as which of those is odious.

Every integer k>=0 corresponds to a unique pair i,j with k = x(i) + y(j), with x(i)=a(i) and y(j)=A380009(j).

Sequences x(n) and y(n) have same growth rate and cross an infinite number of times.

Coordinate pairs (i,j), define a Morton space-filling curve, similar to Z-order curve.

Links

Table of n, a(n) for n=0..63.

Luis Rato, Plot of an NZ-order curve, containing the integers from 0 to 255.

Wikipedia, Morton code map, also known as Z-order curve.

Example

Considering the representation in base 4,
For n=11 = 1011_binary, a(11) -> 1021_base4 -> 2012_base4 = 134.
For n=12 = 1100_binary, a(12) -> 1200_base4 -> 2100_base4 = 144.
Considering all numbers are decomposed in binary, with exponents belonging to odious numbers: 1, 2, 4, 7,...
The sequence of terms together with their binary representation begins:
 n    a(n)      a(n)_bin
 0     0:         0 ~               0
 1     2:        10 ~             2^1
 2     4:       100 ~         2^2
 3     6:       110 ~         2^2+2^1
 4    16:     10000 ~     2^4
 5    18:     10010 ~     2^4   +2^1
 6    20:     10100 ~     2^4+2^2
 7    22:     10110 ~     2^4+2^2+2^1
 8   128:  10000000 ~ 2^7
 9   130:  10000010 ~ 2^7        +2^1
10   132:  10000100 ~ 2^7    +2^2
11   134:  10000110 ~ 2^7    +2^2+2^1
12   144:  10010000 ~ 2^7+2^4

Prog

(PARI) a(n) = { my (v = 0, e); while (n, n -= 2^e = exponent(n); v += 2^(2*e + if (hammingweight(e)%2, 0, 1)); ); return (v); } \\ Rémy Sigrist, Feb 02 2025
(PARI) isok(t) = my(b=Vecrev(binary(t))); for (i=1, #b, if (b[i] && !(hammingweight(i-1)%2), return(0))); return(1); \\ Michel Marcus, Feb 10 2025

Crossrefs

Cf. A000069, A001285, A010059, A380009.

Sequence in context: A341668 A326782 A381957 * A358126 A379291 A096174

Adjacent sequences: A380005 A380006 A380007 * A380009 A380010 A380011

Keyword

base,easy,nonn

Author

Luis Rato, Jan 08 2025

Status

approved