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A271639 Orphans: integers without ancestors, in the sense explained below.

%I A271639 #32 Dec 10 2024 20:25:09
%S A271639 648,649,659,737,738,739,747,748,749,758,759,769,828,829,837,838,839,
%T A271639 846,847,848,849,857,858,859,868,869,879,919,928,929,937,938,939,946,
%U A271639 947,948,949,956,957,958,959,967,968,969,978,979,989,1648,1649,1659,1737,1738
%N A271639 Orphans: integers without ancestors, in the sense explained below.
%C A271639 Look at
%C A271639 2.0.1.6
%C A271639 .2.1.5
%C A271639 We see that 2016 produces 215 if we consider the successive absolute differences of 2016's digits. We call 2016 an "ancestor" of 215. Some integers have many ancestors -- 215 has 28, for example -- and some, the "orphans", have none. The smallest is 648, which is therefore the initial term.
%C A271639 Also numbers that do not appear in A040115. - _Rémy Sigrist_, Jun 10 2017
%C A271639 If n is in the sequence, then so are all numbers that start or end with n or are obtained from n by inserting zeros. - _Robert Israel_, May 27 2019
%C A271639 Eventually almost all numbers are orphans, because there are some impossible substrings, like 919, and any number containing the bad substring is also an orphan. And the fraction of numbers containing any single substring rises asymptotically to 1 (albeit usually slowly). - _Allan C. Wechsler_, Oct 31 2019.
%H A271639 Robert Israel, <a href="/A271639/b271639.txt">Table of n, a(n) for n = 1..10000</a>
%p A271639 filter:= proc(n) local t,L,i;
%p A271639      L:= convert(n,base,10);
%p A271639      t:= {$1..9};
%p A271639      for i from 1 to nops(L) do
%p A271639        t:= select(d -> d >= 0 and d <= 9, map(d -> (d+L[i],d-L[i]), t));
%p A271639        if t = {} then return true fi
%p A271639      od;
%p A271639 false
%p A271639 end proc:
%p A271639 select(filter, [$1..2000]); # _Robert Israel_, May 27 2019
%o A271639 (PARI) \\ Needs PARI from A327270.
%o A271639 select(k->A327270(k)<0, [1..1800]) \\ _Andrew Howroyd_, Dec 10 2024
%Y A271639 Cf. A040115.
%K A271639 nonn,base
%O A271639 1,1
%A A271639 _Eric Angelini_ and _Jean-Marc Falcoz_, Apr 11 2016