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

1,4

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

Prog

(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

Crossrefs

Cf. A005837, A248641, A309890.

Sequence in context: A325028 A073203 A073204 * A248641 A282934 A273619

Adjacent sequences: A322283 A322284 A322285 * A322287 A322288 A322289

Keyword

nonn,easy

Author

Sébastien Palcoux, Aug 28 2019

Status

approved