A005838 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 6. (Formerly M0516)

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 47, 49, 50, 51, 52, 59, 60, 62, 63, 64, 65, 66, 68, 69, 71, 73, 77, 85, 87, 88, 89, 90, 91, 93, 96, 97, 98, 99, 100, 103, 104, 107, 111, 114, 115, 117, 118, 120

Offset

1,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp. 33 (1979), 1353-1359.

Formula

a(n) = A020656(n) + 1.

Maple

N:= 100: # to get a(1)..a(N)
A:= Vector(N):
A[1..5]:= <($1..5)>:
forbid:= {6}:
for n from 6 to N do
  c:= min({$A[n-1]+1::max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[4..n-1], set);
  if ds = {} then next fi;
  ds:= ds intersect convert(map(t -> (c-t)/4, A[1..n-4]), set);
  if ds = {} then next fi;
  ds:= ds intersect convert(map(t -> (c-t)/3, A[2..n-3]), set);
  if ds = {} then next fi;
  ds:= ds intersect convert(map(t -> (c-t)/2, A[3..n-2]), set);
  forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
convert(A, list); # Robert Israel, Jan 04 2016

Prog

(PARI) A005838(n, show=1, i=1, o=6, u=0)={for(n=1, n, show&&print1(i, ", "); u+=1<<i; while(i++, for(s=1, (i-1)\(o-1), for(j=1, o-1, bittest(u, i-s*j)||next(2)); next(2)); next(2))); i} \\ M. F. Hasler, Jan 03 2016

Crossrefs

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):

3-term AP: A005836 (>=0), A003278 (>0);

4-term AP: A005839 (>=0), A005837 (>0);

5-term AP: A020654 (>=0), A020655 (>0);

6-term AP: A020656 (>=0), A005838 (>0);

7-term AP: A020657 (>=0), A020658 (>0);

8-term AP: A020659 (>=0), A020660 (>0);

9-term AP: A020661 (>=0), A020662 (>0);

10-term AP: A020663 (>=0), A020664 (>0).

Sequence in context: A032797 A001967 A226066 * A184486 A047203 A080919

Adjacent sequences: A005835 A005836 A005837 * A005839 A005840 A005841

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

Name and links/references edited by M. F. Hasler, Jan 03 2016

Further edited by N. J. A. Sloane, Jan 04 2016

Status

approved