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A020657 Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 7.

%I A020657 #49 May 06 2025 15:15:30
%S A020657 0,1,2,3,4,5,7,8,9,10,11,12,14,15,16,17,18,19,21,22,23,24,25,26,28,29,
%T A020657 30,31,32,33,35,36,37,38,39,40,49,50,51,52,53,54,56,57,58,59,60,61,63,
%U A020657 64,65,66,67,68,70,71,72,73,74,75,77,78,79,80,81,82,84,85
%N A020657 Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 7.
%C A020657 Also the set of numbers with no "6" in their base-7 representation; see Gerver-Ramsey, also comments in A020654. - _Nathaniel Johnston_, Jun 27 2011
%C A020657 Up to the offset, identical to A037470. There are lexicographically earlier, but non-monotonic sequences which do not contain a 7-term AP, e.g., starting with 0,0,0,0,0,0,1,0,... - _M. F. Hasler_, Oct 05 2014
%H A020657 Nathaniel Johnston, <a href="/A020657/b020657.txt">Table of n, a(n) for n = 1..10000</a>
%H A020657 J. L. Gerver and L. T. Ramsey, <a href="http://dx.doi.org/10.1090/S0025-5718-1979-0537982-0">Sets of integers with no long arithmetic progressions generated by the greedy algorithm</a>, Math. Comp., 33 (1979), 1353-1359.
%p A020657 seq(`if`(numboccur(6,convert(n,base,7))=0,n,NULL),n=0..85); # _Nathaniel Johnston_, Jun 27 2011
%t A020657 Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] (* _Jean-François Alcover_, Jan 27 2023 *)
%o A020657 (PARI) a(n)=vector(#n=digits(n-1, 6), i, 7^(#n-i))*n~ \\ _M. F. Hasler_, Oct 05 2014
%o A020657 (Python)
%o A020657 from gmpy2 import digits
%o A020657 def A020657(n): return int(digits(n-1,6),7) # _Chai Wah Wu_, May 06 2025
%Y A020657 Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
%Y A020657 3-term AP: A005836 (>=0), A003278 (>0);
%Y A020657 4-term AP: A005839 (>=0), A005837 (>0);
%Y A020657 5-term AP: A020654 (>=0), A020655 (>0);
%Y A020657 6-term AP: A020656 (>=0), A005838 (>0);
%Y A020657 7-term AP: A020657 (>=0), A020658 (>0);
%Y A020657 8-term AP: A020659 (>=0), A020660 (>0);
%Y A020657 9-term AP: A020661 (>=0), A020662 (>0);
%Y A020657 10-term AP: A020663 (>=0), A020664 (>0).
%K A020657 nonn,easy
%O A020657 1,3
%A A020657 _David W. Wilson_
%E A020657 Name edited by _M. F. Hasler_, Oct 10 2014. Further edited by _N. J. A. Sloane_, Jan 04 2016