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

%I A380009 #52 Aug 11 2025 10:25:19
%S A380009 0,1,8,9,32,33,40,41,64,65,72,73,96,97,104,105,512,513,520,521,544,
%T A380009 545,552,553,576,577,584,585,608,609,616,617,1024,1025,1032,1033,1056,
%U A380009 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
%N A380009 Numbers t whose binary expansion Sum 2^e_i has exponents e_i which are evil numbers (A001969).
%C A380009 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.
%C A380009 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.
%C A380009 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.
%C A380009 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).
%C A380009 Sequences x(n) and y(n) have same growth rate and cross an infinite number of times.
%C A380009 Coordinate pairs (i,j), define a Morton space-filling curve, similar to Z-order curve.
%H A380009 Luis Rato, <a href="/A380009/a380009.png">Plot of an NZ-order curve</a>, containing the integers from 0 to 255.
%H A380009 Wikipedia, Morton code map, also known as <a href="https://en.wikipedia.org/wiki/Z-order_curve">Z-order curve</a>.
%e A380009 For n=11 = 1011_binary, a(11) = 1021_base4 = 41.
%e A380009 All numbers are also decomposed in binary, with exponents belonging to evil numbers: 0, 3, 5, 6, ...
%e A380009 The sequence of terms begins:
%e A380009  n    a(n)      a(n)_bin
%e A380009  0     0:         0 ~               0
%e A380009  1     1:         1 ~             2^0
%e A380009  2     8:      1000 ~         2^3
%e A380009  3     9:      1001 ~         2^3+2^0
%e A380009  4    32:    100000 ~     2^5
%e A380009  5    33:    100001 ~     2^5    +2^0
%e A380009  6    40:    101000 ~     2^5+2^3
%e A380009  7    41:    101001 ~     2^5+2^3+2^0
%e A380009  8    64:   1000000 ~ 2^6
%e A380009  9    65:   1000001 ~ 2^6        +2^0
%e A380009 10    72:   1001000 ~ 2^6    +2^3
%e A380009 11    73:   1001001 ~ 2^6    +2^3+2^0
%o A380009 (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
%Y A380009 Cf. A000069, A001285, A001969, A010059, A380008.
%K A380009 base,easy,nonn
%O A380009 0,3
%A A380009 _Luis Rato_, Jan 09 2025