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A005838
Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 6.
(Formerly M0516)
28
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
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
KEYWORD
nonn
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