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A020654
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Lexicographically earliest infinite increasing sequence of nonnegative numbers containing no 5-term arithmetic progression.
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49
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0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 125, 126, 127
(list;
graph;
refs;
listen;
history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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This is also the set of numbers with no "4" in their base-5 representation. In fact, for any prime p, the sequence consisting of numbers with no (p-1) in their base-p expansion is the same as the earliest sequence containing no p-term arithmetic progression. - Nathaniel Johnston, Jun 26-27 2011
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LINKS
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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.
Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.
Index entries for 5-automatic sequences.
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MAPLE
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seq(`if`(numboccur(4, convert(n, base, 5))=0, n, NULL), n=0..127); # Nathaniel Johnston, Jun 27 2011
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MATHEMATICA
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Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 5 ], 4 ]==0)& ]
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PROG
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(PARI) is(n)=while(n>4, if(n%5==4, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017
(Python)
from sympy.ntheory.factor_ import digits
print([n for n in range(201) if digits(n, 5)[1:].count(4)==0]) # Indranil Ghosh, May 23 2017
(Julia)
function a(n)
m, r, b = n, 0, 1
while m > 0
m, q = divrem(m, 4)
r += b * q
b *= 5
end
r end; [a(n) for n in 0:66] |> println # Peter Luschny, Jan 03 2021
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CROSSREFS
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Cf. A023717.
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: A087069 A023737 A037459 * A182777 A214988 A028804
Adjacent sequences: A020651 A020652 A020653 * A020655 A020656 A020657
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KEYWORD
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nonn,easy,changed
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019
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STATUS
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approved
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