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

Author

David W. Wilson

Keywords

• nonn
• easy

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.

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).

Programs

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

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 *)

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

Python

from gmpy2 import digits
def A020654(n): return int(digits(n-1,4),5) # Chai Wah Wu, May 06 2025

Formula

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

Extensions

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