A020658 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 7.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 99

Offset

1,2

Comments

This is different from A047304: note the gap between 41 and 50. - M. F. Hasler, Oct 07 2014

Links

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

Formula

a(n) = A020657(n)+1. - M. F. Hasler, Oct 07 2014

Maple

Noap:= proc(N, m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A, forbid, n, c, ds, j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]), set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]), set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
convert(A, list)
end proc:
Noap(100, 7); # Robert Israel, Jan 04 2016

Mathematica

Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] + 1 (* Jean-François Alcover, Aug 18 2023, after M. F. Hasler *)

Crossrefs

Cf. A047304.

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: A187396 A020659 A047304 * A195178 A043092 A023801

Adjacent sequences: A020655 A020656 A020657 * A020659 A020660 A020661

Keyword

nonn

Author

David W. Wilson

Status

approved