A020654 - OEIS

%I #45 Jan 03 2021 14:35:53

%S 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,

%T 36,37,38,40,41,42,43,50,51,52,53,55,56,57,58,60,61,62,63,65,66,67,68,

%U 75,76,77,78,80,81,82,83,85,86,87,88,90,91,92,93,125,126,127

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

%C 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

%H Nathaniel Johnston, <a href="/A020654/b020654.txt">Table of n, a(n) for n = 1..10000</a>

%H 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.

%H Samuel S. Wagstaff, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0325500-5">On k-free sequences of integers</a>, Math. Comp., 26 (1972), 767-771.

%H <a href="/index/Ar#5-automatic">Index entries for 5-automatic sequences</a>.

%p seq(`if`(numboccur(4,convert(n,base,5))=0,n,NULL),n=0..127); # _Nathaniel Johnston_, Jun 27 2011

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

%o (PARI) is(n)=while(n>4, if(n%5==4, return(0)); n\=5); 1 \\ _Charles R Greathouse IV_, Feb 12 2017

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o print([n for n in range(201) if digits(n, 5)[1:].count(4)==0]) # _Indranil Ghosh_, May 23 2017

%o (Julia)

%o function a(n)

%o     m, r, b = n, 0, 1

%o     while m > 0

%o         m, q = divrem(m, 4)

%o         r += b * q

%o         b *= 5

%o     end

%o r end; [a(n) for n in 0:66] |> println # _Peter Luschny_, Jan 03 2021

%Y Cf. A023717.

%Y Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):

%Y 3-term AP: A005836 (>=0), A003278 (>0);

%Y 4-term AP: A005839 (>=0), A005837 (>0);

%Y 5-term AP: A020654 (>=0), A020655 (>0);

%Y 6-term AP: A020656 (>=0), A005838 (>0);

%Y 7-term AP: A020657 (>=0), A020658 (>0);

%Y 8-term AP: A020659 (>=0), A020660 (>0);

%Y 9-term AP: A020661 (>=0), A020662 (>0);

%Y 10-term AP: A020663 (>=0), A020664 (>0).

%K nonn,easy

%O 1,3

%A _David W. Wilson_

%E Added "infinite" to definition. - _N. J. A. Sloane_, Sep 28 2019