A322286 Lexicographically earliest sequence of positive integers without 4 terms in a weakly increasing arithmetic progression.

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 1, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 3, 5, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 3, 4, 2, 3, 2, 2, 2, 3, 3, 1, 3, 3, 3, 5, 5, 4, 1, 1, 1, 3, 1, 2, 3, 1, 5, 3, 2, 6, 1, 3, 2, 1, 3, 2, 1, 1, 3, 3, 1, 1, 1

Offset: 1

Author

Sébastien Palcoux, Aug 28 2019

Keywords

• nonn
• easy

Comments

This is a variation of A248641 (where we only exclude weakly increasing arithmetic progressions): they differ from the 101st term.

It is also a variation of A309890 where 3-term is replaced by 4-term.

The numbers n for which the n-th term is 1 are given by A005837.

There is no upper bound, because if there were an upper bound r then there must be s <= r such that the set of numbers n for which the n-th term is s has positive density and this contradicts Szemerédi's theorem.

Assuming Erdős's conjecture on arithmetic progressions, for a fixed positive integer r, the sum of the reciprocals of the numbers n for which the n-th term is r converges.

Links

• Sébastien Palcoux, Table of n, a(n) for n = 1..10000
• Wikipedia, Szemerédi's theorem
• Wikipedia, Erdős conjecture on arithmetic progressions

Crossrefs

Cf. A005837, A248641, A309890.

Programs

SageMath

cpdef FourFree(int n):
   cdef int i,r,k,s,L1,L2,L3
   cdef list L,Lb
   cdef set b
   L=[1,1,1]
   for k in range(3,n):
      b=set()
      for i in range(k):
         if 3*((k-i)/3)==k-i:
            r=(k-i)/3
            L1,L2,L3=L[i],L[i+r],L[i+2*r]
            s=3*(L2-L1)+L1
            if s>0 and L3==2*(L2-L1)+L1:
               if L1<=L2:
                  b.add(s)
      if 1 not in b:
         L.append(1)
      else:
         Lb=list(b)
         Lb.sort()
         for t in Lb:
            if t+1 not in b:
               L.append(t+1)
               break
   return L