Count my odd digits, please

NAME

Lexicographically earliest sequence of distinct terms > 0 such that the number of odd digits present in each term is described by another term, or itself if possible (see how in the Comments section).

DATA

20, 40, 10, 60, 31, 80, 52, 200, 71, 92, 101, 112, 122, 131, 143, 152, 162, 171, 183, 220, 192, 240, 212, 260, 231, 280, 252, 400, 271, 292, 301, 312, 322, 331, 343, 352, 362, 371, 383, 420, 392, 440, 412, 460, 431, 480, 452, 600, 471, 492, 501, 512, 522, 531, 543, 552, …

COMMENTS

All integers N of the sequence are the concatenation of two parts, X and Y. 

X is the position of N in the sequence;

Y is the number of odd digits present in N. 

The term a(5) = 31 means that the 3rd term of the sequence contains 1 odd digit: this is true as a(3) = 10. 

The first term a(1) = 20 means that the 2nd term contains 0 odd digit: this is true as a(2) = 40. Etc. 

____________________

Possible variants of this sequence might be:

NAME = Lexicographically earliest sequence of distinct terms > 0 such that the number of even digits present in each term is described by another term, or itself if possible (underlined):

DATA = 20, 31, 12, 42, 60, 71, 52, 82, 100, 111, 92, 121, 140, 151, 161, 132, 180, 197...

NAME = Lexicographically earliest sequence of distinct terms > 0 such that the number of prime digits (2, 3, 5, 7) present in each term is described by another term, or itself if possible (underlined):

DATA = 10, 21, 31, 40, 51, 60, 71, 80, 90, 100, 110, 121, 131, 140, 151, 160, 171, 180, 190, 201, 211, 223, 233, 241, 253, 261, 273, 281, 291, 301, 311, 323, 333, 342, 353, 361, 373, 381, 391, 400, 410, 421, 431, 440, 451, 460, …

NAME = Lexicographically earliest sequence of distinct terms > 0 such that the number of nonprime digits (0, 1, 4, 6, 8, 9) present in each term is described by another term, or itself if possible (underlined):

DATA = 20, 32, 11, 50, 72, 41, 60, 90, 80, 102, 112, 130, 222, 122, 141, 153, 161, 173, 181, 193, 202, 211, 231, 250, 272, 241, 280, 322, 262, 291, 302, 311, 331, ...

NAME = Lexicographically earliest sequence of distinct terms > 0 such that the number of Fibonacci digits (0, 1, 2, 3, 5, 8) present in each term is described by another term, or itself if possible (underlined):

DATA = 12, 22,  32, 41, 52, 61, 71, 82, 91, 103, 113, 123, 133, 142, 153, 162, 172, 183, 192, 203, 213, 223, 242, 253, 262, 272, 283, 292, 303, 313, 323, 333, 342, 353, 362, 372, 383, 392, 402, 412, 422, 432, 441, 452, 461, 471, 482, 491, 503, 513, 523, 533, 542, 553, 562, 572, 583, 592, 602, 612, 632, 641, 652, 661, 671, 682, 691, 702, ...

NAME = Lexicographically earliest sequence of distinct terms > 0 such that the number of non-Fibonacci digits (4, 6, 7, 9) present in each term is described by another term, or itself if possible (underlined):

DATA = 10, 20, 30, 41, 50, 61, 71, 80, 91, 100, 110, 120, 130, 141, 150, 161, 171, 180, 191, 200, 210, 220, 230, 241, 250, 261, 271, 280, 291, 300, 310, 320, 330, 341, 350, 361, 371, 380, 391, 401, 411, 421, 431, 442, 451, 462, 472, 481, 492, 500, 510, 520, 530, 541, 550, 561, 571, 580, 591, 601, 611, 621, 631, 642, 651, 662, ...