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

Author

Luis Rato, Jan 08 2025

Keywords

• base
• easy
• nonn

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.

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A000069, A001285, A010059, A380009.

Programs

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