

A020654


Lexicographically earliest infinite increasing sequence of nonnegative numbers containing no 5term arithmetic progression.


49



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
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OFFSET

1,3


COMMENTS

This is also the set of numbers with no "4" in their base5 representation. In fact, for any prime p, the sequence consisting of numbers with no (p1) in their basep expansion is the same as the earliest sequence containing no pterm arithmetic progression.  Nathaniel Johnston, Jun 2627 2011


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), 13531359.
Samuel S. Wagstaff, Jr., On kfree sequences of integers, Math. Comp., 26 (1972), 767771.
Index entries for 5automatic sequences.


MAPLE

seq(`if`(numboccur(4, convert(n, base, 5))=0, n, NULL), n=0..127); # Nathaniel Johnston, Jun 27 2011


MATHEMATICA

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 5 ], 4 ]==0)& ]


PROG

(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


CROSSREFS

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):
3term AP: A005836 (>=0), A003278 (>0);
4term AP: A005839 (>=0), A005837 (>0);
5term AP: A020654 (>=0), A020655 (>0);
6term AP: A020656 (>=0), A005838 (>0);
7term AP: A020657 (>=0), A020658 (>0);
8term AP: A020659 (>=0), A020660 (>0);
9term AP: A020661 (>=0), A020662 (>0);
10term AP: A020663 (>=0), A020664 (>0).
Sequence in context: A087069 A023737 A037459 * A182777 A214988 A028804
Adjacent sequences: A020651 A020652 A020653 * A020655 A020656 A020657


KEYWORD

nonn,easy,changed


AUTHOR

David W. Wilson


EXTENSIONS

Added "infinite" to definition.  N. J. A. Sloane, Sep 28 2019


STATUS

approved



