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A356675 Lexicographically earliest infinite sequence satisfying a(1) > -1 and a(n-1) = A075826(a(n)).

%I A356675 #47 Nov 26 2022 09:38:28
%S A356675 1,5,9,16,27,38,48,58,66,76,87,98,117,136,155,177,198,215,235,254,275,
%T A356675 295,310,333,350,372,394,411,433,452,474,495,514,535,555,576,598,615,
%U A356675 635,650,669,689,705,728,749,773,795,810,833,850,872,894,913,934,950,973,994,1013,1034,1050,1071,1093
%N A356675 Lexicographically earliest infinite sequence satisfying a(1) > -1 and a(n-1) = A075826(a(n)).
%C A356675 There is at least one infinite sequence that satisfies the parameters of the sequence, because it is always true that A075826(k) < k for any k. (For the statement to be false, the number of letters must be less than or equal to 0.)
%C A356675 All known values in this sequence are guaranteed to be in any infinite sequence which can be produced, since iterating A075826(k) where the starting value of k is any number greater than 11126 converges at 6890 or larger. This proves any infinite sequence must contain 6890 and all values returned by iterating A075826 down to 0.
%H A356675 Michel Marcus, <a href="/A356675/b356675.txt">Table of n, a(n) for n = 1..11623</a>
%F A356675 a(n-1) = A075826(a(n)).
%e A356675 A075826 yields the number of letters in US English spelling of each number subtracted from the number's value. For example, A075826(5) yields 1, because 5 - 4 (F-I-V-E) is 1. Our formula reveals that if 5 is in our sequence, it must come immediately after 1, which it does. However, 4, which must come immediately after 0, is not in our sequence, which we can prove because there exists no number k such that A075826(k) = 4. Each number less than the largest known value in this sequence produces a finite sequence unless it is in this sequence, because some number eventually is a dead end like 4.
%o A356675 (PARI) f(k) = k - A005589(k);
%o A356675 listd(nn) = {nn *= 2; my(vs = vector(nn)); my(list = List()); my(m=1, lbound = 0); listput(list, m); while (m < nn, if (vs[m] == 0, vs[m] = Vec(select(x->(x==m), vector(100, k, f(k+m-1)), 1)); if (#vs[m], vs[m] = apply(x->(x+m-1), vs[m]))); my(ok = 1, vc = vs[m]); if (! #vc, ok = 0, vc = select(x->(x>lbound), vc); if (! #vc, ok = 0);); if (!ok, lbound = list[#list]; listpop(list); if (! #list, return()); m = list[#list];, lbound = 0; m = vc[1]; listput(list, m););); Vec(select(x->(x<=nn/2), list));} \\ _Michel Marcus_, Aug 31 2022
%Y A356675 Cf. A005589, A075826.
%K A356675 nonn,word
%O A356675 1,2
%A A356675 _Aidan Clarke_, Aug 22 2022