A367065 a(1)=2, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == 2 (mod n+2).

2, 4, 1, 7, 9, 3, 12, 14, 5, 17, 6, 20, 22, 8, 25, 27, 10, 30, 11, 33, 35, 13, 38, 40, 15, 43, 16, 46, 48, 18, 51, 19, 54, 56, 21, 59, 61, 23, 64, 24, 67, 69, 26, 72, 74, 28, 77, 29, 80, 82, 31, 85, 32, 88, 90, 34, 93, 95, 36, 98, 37, 101, 103, 39, 106, 108, 41, 111, 42, 114

Offset: 1

Author

Zenan Sabanac, Nov 03 2023

Keywords

• nonn

Comments

This is the Avdispahić-Zejnulahi sequence AZ(2). For a positive integer k, the Avdispahić-Zejnulahi sequence AZ(k) is given by: a(1)=k, thereafter a(n) is the least positive integer not yet in the sequence such that Sum_{i=1..n} a(i) == k (mod n+k). It is interesting to note that (AZ(k)) represents a sequence of permutations of the set of positive integers. (See Links section for details concerning AZ(1).)

Links

• Muharem Avdispahić and Faruk Zejnulahi, An integer sequence with a divisibility property, Fibonacci Quarterly, Vol. 58:4 (2020), 321-333.
• Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], 2023.

Crossrefs

A340510 is the AZ(1) sequence. A002251 is the AZ(0) sequence.

Programs

Mathematica

lst={2}; f[s_List]:=Block[{k=1,len=3+Length@lst,t=Plus@@lst},While[MemberQ[s,k]||Mod[k+t,len]!=2,k++]; AppendTo[lst,k]]; Nest[f,lst,100]

Python

z_list = [-1, 2, 4]
m_list = [-1, 0, 1]
n = 2
for n in range(2, 100):
    if m_list[n] in z_list:
        m_list.append(m_list[n] + 1)
        z_list.append(m_list[n+1] + n+2)
    else:
        m_list.append(m_list[n])
        z_list.append(m_list[n+1])
print(z_list[1:])