cp's OEIS Frontend

A329448 Numbers n which appear twice in the sequence starting with n and subsequent terms given by 10c + d where c is the number of times (> 0) the least frequent digit d has appeared so far, smallest d in case of tie.

%I A329448 #19 Nov 16 2019 02:29:13
%S A329448 10,12,13,14,15,16,17,18,19,20,22,23,24,25,26,27,28,29,30,34,35,36,37,
%T A329448 38,39,40,44,45,46,47,48,49,50,55,56,57,58,59,60,66,67,68,69,70,77,78,
%U A329448 79,80,88,89,90,99,100,102,103,104,105,106,107,108,109,115,122,123,124,125,126,127,128
%N A329448 Numbers n which appear twice in the sequence starting with n and subsequent terms given by 10c + d where c is the number of times (> 0) the least frequent digit d has appeared so far, smallest d in case of tie.
%C A329448 Sequence A329447 is defined as above with n = 0, but one can take any other initial term n. This sequence lists those n which reappear later in the resulting sequence. To know whether n reappears, it is sufficient to compute the sequence up to the point where the last digit of n has appeared at least n/10 times. (The sequence may be not strictly increasing, but each subsequence of terms ending in the same digit is, ignoring the initial term.)
%C A329448 There are 369 integers < 1000 and 2696 integers < 10^4 in this sequence.
%C A329448 All powers of 10 appear to be in the sequence, but this is only verified up to 10^7, not proved in general.
%H A329448 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2019/11/look-left-and-say.html">Look left and say</a>, Nov 14 2019.
%e A329448 Single-digit numbers cannot reappear: all terms are of the form 10c + d, c > 0.
%e A329448 10 does reappear, because after this initial value, the next term is 10 for "one '1' [appeared so far]".
%e A329448 11 does not reappear, because after this initial value, the digit 1 has appeared already twice, and a term 11 would mean that only "one 1" would have appeared so far.
%e A329448 The numbers n = 12, ..., 19 reappear because they are, as initial term, followed by 11 for "one '1' [has appeared so far]"; thereafter the second digit of n will be the least frequent one, having appeared once, so the next term will be n for "one (second digit of n) [has appeared so far]".
%e A329448 The number 20 reappears because it is followed by 10 (one '0'), then 11 (one '1'), then 12 (one '2'), and then 20 (two '0's).
%e A329448 The number 21 does not reappear because it is followed by 11 (one '1'); at this point three '1's have already appeared and we can never again have 21 meaning that only "two '1's" have appeared so far.
%o A329448 (PARI) select( is(n)={my(c=Vec(0,10),t=-n,i, n0=n%10+1, n1=n\10); while(t!=n&&c[n0]<=n1, for(j=1,#i=vecsort(c,,1), if(c[i[j]], t=c[i[j]]*10+i[j]-1; break)); for(j=1, #i=digits(t), c[i[j]+1]++));t==n&&n}, [0..130])
%Y A329448 Cf. A329447.
%K A329448 nonn,base
%O A329448 1,1
%A A329448 _Eric Angelini_ and _M. F. Hasler_, Nov 14 2019