%I #34 Aug 19 2023 19:04:02

%S 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,

%T 30,31,32,33,34,36,37,38,39,40,41,50,51,52,53,54,55,57,58,59,60,61,62,

%U 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

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

%C This is different from A047304: note the gap between 41 and 50. - _M. F. Hasler_, Oct 07 2014

%H Robert Israel, <a href="/A020658/b020658.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A020657(n)+1. - _M. F. Hasler_, Oct 07 2014

%p Noap:= proc(N,m)

%p # N terms of earliest increasing seq with no m-term arithmetic progression

%p local A,forbid,n,c,ds,j;

%p A:= Vector(N):

%p A[1..m-1]:= <($1..m-1)>:

%p forbid:= {m}:

%p for n from m to N do

%p c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);

%p A[n]:= c;

%p ds:= convert(map(t -> c-t, A[m-2..n-1]),set);

%p for j from m-2 to 2 by -1 do

%p ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);

%p if ds = {} then break fi;

%p od;

%p forbid:= select(`>`,forbid,c) union map(`+`,ds,c);

%p od:

%p convert(A,list)

%p end proc:

%p Noap(100, 7); # _Robert Israel_, Jan 04 2016

%t Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] + 1 (* _Jean-François Alcover_, Aug 18 2023, after _M. F. Hasler_ *)

%Y Cf. A047304.

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

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

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

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

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

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

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

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

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

%K nonn

%O 1,2

%A _David W. Wilson_