A380009 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are evil numbers (A001969).

0, 1, 8, 9, 32, 33, 40, 41, 64, 65, 72, 73, 96, 97, 104, 105, 512, 513, 520, 521, 544, 545, 552, 553, 576, 577, 584, 585, 608, 609, 616, 617, 1024, 1025, 1032, 1033, 1056, 1057, 1064, 1065, 1088, 1089, 1096, 1097, 1120, 1121, 1128, 1129, 1536, 1537, 1544, 1545, 1568, 1569, 1576, 1577, 1600, 1601, 1608, 1609, 1632, 1633, 1640, 1641

Offset: 0

Author

Luis Rato, Jan 09 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.

Any pair 2*p and 2*p+1 has one evil and the other odious, 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 evil.

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)=A380008(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

			For n=11 = 1011_binary, a(11) = 1021_base4 = 41.
All numbers are also decomposed in binary, with exponents belonging to evil numbers: 0, 3, 5, 6, ...
The sequence of terms begins:
 n    a(n)      a(n)_bin
 0     0:         0 ~               0
 1     1:         1 ~             2^0
 2     8:      1000 ~         2^3
 3     9:      1001 ~         2^3+2^0
 4    32:    100000 ~     2^5
 5    33:    100001 ~     2^5    +2^0
 6    40:    101000 ~     2^5+2^3
 7    41:    101001 ~     2^5+2^3+2^0
 8    64:   1000000 ~ 2^6
 9    65:   1000001 ~ 2^6        +2^0
10    72:   1001000 ~ 2^6    +2^3
11    73:   1001001 ~ 2^6    +2^3+2^0
		

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, A001969, A010059, A380008.

Programs

PARI

isok(k) = my(b=binary(k), v=apply(x->#b-x, Vec(select(x->x, b, 1)))); #v == #select(x->(hammingweight(x)%2==0), v); \\ Michel Marcus, Jan 11 2025