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

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, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

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

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

Links

• Nathaniel Johnston, 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.

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).

Programs

Maple

seq(`if`(numboccur(6,convert(n,base,7))=0,n,NULL),n=0..85); # Nathaniel Johnston, Jun 27 2011

Mathematica

Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] (* Jean-François Alcover, Jan 27 2023 *)

PARI

a(n)=vector(#n=digits(n-1, 6), i, 7^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

Python

from gmpy2 import digits
def A020657(n): return int(digits(n-1,6),7) # Chai Wah Wu, May 06 2025

Extensions

Name edited by M. F. Hasler, Oct 10 2014. Further edited by N. J. A. Sloane, Jan 04 2016