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A367067 a(1)=3, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == 3 (mod n+3).

%I A367067 #20 Dec 16 2023 14:16:09
%S A367067 3,5,1,8,2,11,13,4,16,18,6,21,7,24,26,9,29,10,32,34,12,37,39,14,42,15,
%T A367067 45,47,17,50,52,19,55,20,58,60,22,63,23,66,68,25,71,73,27,76,28,79,81,
%U A367067 30,84,31,87,89,33,92,94,35,97,36,100,102,38,105
%N A367067 a(1)=3, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == 3 (mod n+3).
%C A367067 This is the Avdispahić-Zejnulahi sequence AZ(3).
%C A367067 Note that AZ(3) is the third term in a sequence of permutations of the set of positive integers defined by a specific divisibility property (see Links section and Crossrefs for details).
%H A367067 Muharem Avdispahić and Faruk Zejnulahi, <a href="https://www.researchgate.net/publication/341726940_AN_INTEGER_SEQUENCE_WITH_A_DIVISIBILITY_PROPERTY">An integer sequence with a divisibility property</a>, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
%H A367067 Jeffrey Shallit, <a href="https://arxiv.org/abs/2308.06544">Proving properties of some greedily-defined integer recurrences via automata theory</a>, arXiv:2308.06544 [cs.DM], 2023.
%t A367067 lst = {3};
%t A367067 f[s_List] := Block[{k = 1, len = 4 + Length@lst, t = Plus @@ lst},
%t A367067   While[MemberQ[s, k] || Mod[k + t, len] != 3, k++];
%t A367067   AppendTo[lst, k]]; Nest[f, lst, 100]
%o A367067 (Python)
%o A367067 z_list=[-1,3,5]
%o A367067 m_list=[-1,0,1]
%o A367067 n=2
%o A367067 for n in range(2, 100):
%o A367067     if m_list[n] in z_list:
%o A367067         m_list.append(m_list[n] + 1)
%o A367067         z_list.append(m_list[n+1] + n+3)
%o A367067     else:
%o A367067         m_list.append(m_list[n])
%o A367067         z_list.append(m_list[n+1])
%o A367067 print(z_list[1:])
%Y A367067 Cf. A340510, A367065.
%K A367067 nonn
%O A367067 1,1
%A A367067 _Zenan Sabanac_, Nov 03 2023