A020654 Lexicographically earliest infinite increasing sequence of nonnegative numbers containing no 5-term arithmetic progression.

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

Offset

1,3

Comments

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

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

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.

Formula

Sum_{n>=2} 1/a(n) = 7.7794910022243020875287956248411192066951785182667316905881486574421016471305408306837031955619272391023... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

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)& ]
Select[Range[0, 120], DigitCount[#, 5, 4] == 0 &] (* Amiram Eldar, Apr 14 2025 *)
A020654[n_] := FromDigits[IntegerString[n - 1, 4], 5];
Array[A020654, 100] (* Paolo Xausa, Jan 22 2026 *)

Prog

(PARI) is(n)=while(n>4, if(n%5==4, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017
(PARI) a(n)= fromdigits(digits(n, 4), 5) \\ Ruud H.G. van Tol, Jan 22 2026
(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
(Python)
from gmpy2 import digits
def A020654(n): return int(digits(n-1, 4), 5) # Chai Wah Wu, May 06 2025
(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):

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

Keyword

nonn,easy,changed

Author

David W. Wilson

Extensions

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

Status

approved