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

648, 649, 659, 737, 738, 739, 747, 748, 749, 758, 759, 769, 828, 829, 837, 838, 839, 846, 847, 848, 849, 857, 858, 859, 868, 869, 879, 919, 928, 929, 937, 938, 939, 946, 947, 948, 949, 956, 957, 958, 959, 967, 968, 969, 978, 979, 989, 1648, 1649, 1659, 1737, 1738

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Apr 11 2016

Keywords

• nonn
• base

Comments

Look at

2.0.1.6

.2.1.5

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.

Also numbers that do not appear in A040115. - Rémy Sigrist, Jun 10 2017

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

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.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A040115.

Programs

Maple

filter:= proc(n) local t,L,i;
     L:= convert(n,base,10);
     t:= {$1..9};
     for i from 1 to nops(L) do
       t:= select(d -> d >= 0 and d <= 9, map(d -> (d+L[i],d-L[i]), t));
       if t = {} then return true fi
     od;
false
end proc:
select(filter, [$1..2000]); # Robert Israel, May 27 2019

PARI

\\ Needs PARI from A327270.
select(k->A327270(k)<0, [1..1800]) \\ Andrew Howroyd, Dec 10 2024