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A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

%I A363186 #17 Feb 08 2024 09:46:24
%S A363186 1,10,98,767,111,122,2,11,100,889,110,4490,400,560,1096,124,20,129,70,
%T A363186 502,93,171,212,361,512,26,21,36,54,14,1011,139,99,59,550,684,345,102,
%U A363186 1021,1999,2871,137,892,89,126,875,510,994,586,2012,662,1836,201,405,388,2007,2798,1641,50,340
%N A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).
%C A363186 In the first 10000 terms the smallest number that has not yet appeared is 696; it is therefore likely all numbers eventually appear although this is unknown.
%H A363186 Scott R. Shannon, <a href="/A363186/b363186.txt">Table of n, a(n) for n = 1..10000</a>
%H A363186 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2023/07/echecs-et-maths.html">Échecs et Maths</a>, Personal blog, July 2023.
%e A363186 a(2) = 10 as a(1) + 10 = 1 + 10 = 11 which is a substring of "1" + "10" = "110".
%e A363186 a(3) = 98 as a(1) + a(2) + 98 = 1 + 10 + 98 = 109 which is a substring of "1" + "10" + "98" = "11098".
%e A363186 a(4) = 767 as a(1) + a(2) + a(3) + 767 = 1 + 10 + 98 + 767 = 876 which is a substring of "1" + "10" + "98" + "767" = "11098767".
%o A363186 (Python)
%o A363186 from itertools import islice
%o A363186 def agen(): # generator of terms
%o A363186     s, mink, aset, concat = 1, 2, {1}, "1"
%o A363186     yield from [1]
%o A363186     while True:
%o A363186         an = mink
%o A363186         while an in aset or not str(s+an) in concat+str(an): an += 1
%o A363186         aset.add(an); s += an; concat += str(an); yield an
%o A363186         while mink in aset: mink += 1
%o A363186 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Feb 08 2024
%Y A363186 Cf. A359482, A300000, A339144, A357082.
%K A363186 nonn,base
%O A363186 1,2
%A A363186 _Scott R. Shannon_ and _Eric Angelini_, Jul 07 2023