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Smallest sequence which lists the position of digits "2" in the sequence.
6

%I #6 Oct 02 2024 06:14:14

%S 3,4,22,30,31,33,34,35,36,37,38,42,43,44,45,52,202,222,223,302,2220,

%T 3000,3200,3300,3301,3303,3304,3305,3306,3307,3308,3309,3310,3311,

%U 3313,3314,3315,3316,3317,3318,3319,3330,3331,3333,3334,3335,3336,3337,3338

%N Smallest sequence which lists the position of digits "2" in the sequence.

%C The lexicographically earliest sequence such that a(1),a(2),a(3),... is the (increasing) list of the positions of digits "1" in the string obtained by concatenating all these terms, written in base 10.

%e We cannot have a(1)=1 (since then there's no 2 in the first place), nor a(1)=2 (since then the first occurrence of a "2" would be at position 1).

%e But a(1)=3 is possible, "predicting" that the first occurrence of a digit "2" will be in the 3rd digit.

%e Then a(2)=4 is the smallest possible choice for a(2).

%e The next two digits (= the 3rd and 4th digit of the sequence) must be a "2", in view of a(1) and a(2). Thus a(3)=22 is the smallest possible choice.

%e This means that the next digit "2" will occur as the 22nd digit of the sequence, so the following terms are the least possible numbers without digit "2": 30,31,33,...,38. These make up digits 5 to 20 of the sequence.

%e The following number must have a "2" as second digit, the smallest possibility is 42.

%o (PARI) concat([ [3,4,22], vector((22-4)/2-1,i,i+30-(i<=2)), vector(4,i,42+i-1), [52,202,222,223,302,2220,3000,3200], select(x -> x%10-2 & x\10%10-2 & x\100%10-2, vector((202-52)\4+13,i,3300+i-1)) ])

%Y Cf. A098645, A167519, A167520.

%K base,nonn,nice

%O 1,1

%A _M. F. Hasler_, Nov 19 2009