Displaying 1-10 of 48 results found.
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
Numbers whose base-3 representation contains no 2.
(Formerly M2353)
+10
239
0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 243, 244, 246, 247, 252, 253, 255, 256, 270, 271, 273, 274, 279, 280, 282, 283, 324, 325, 327, 328, 333, 334, 336, 337, 351, 352
COMMENTS
3 does not divide binomial(2s, s) if and only if s is a member of this sequence, where binomial(2s, s) = A000984(s) are the central binomial coefficients. This is the lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 3. - Robert Craigen (craigenr(AT)cc.umanitoba.ca), Jan 29 2001
Also final value of n - 1 written in base 2 and then read in base 3 and with finally the result translated in base 10. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
Also numbers such that the balanced ternary representation is the same as the base 3 representation. - Alonso del Arte, Feb 25 2011 Fixed point of the morphism: 0 -> 01; 1 -> 34; 2 -> 67; ...; n -> (3n)(3n+1), starting from a(1) = 0. - Philippe Deléham, Oct 22 2011 It appears that this sequence lists the values of n which satisfy the condition sum(binomial(n, k)^(2*j), k = 0..n) mod 3 <> 0, for any j, with offset 0. See Maple code. - Gary Detlefs, Nov 28 2011 Also, it follows from the above comment by Philippe Lallouet that the sequence must be generated by the rules: a(1) = 0, and if m is in the sequence then so are 3*m and 3*m + 1. - L. Edson Jeffery, Nov 20 2015
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E10, pp. 317-323.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023). Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019. Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018. P. Erdős, V. Lev, G. Rauzy, C. Sandor, and A. Sarkozy, Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., Vol. 200, No. 1-3 (1999), pp. 119-135 (see Table 1). alternate link. A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978, remark 1 ( PDF, PS, TeX).
FORMULA
Numbers n such that the coefficient of x^n is > 0 in prod (k >= 0, 1 + x^(3^k)). - Benoit Cloitre, Jul 29 2003 a(n+1) = Sum_{k=0..m} b(k)* 3^k and n = Sum( b(k)* 2^k ).
a(2n+1) = 3a(n+1), a(2n+2) = a(2n+1) + 1, a(0) = 0.
a(n+1) = 3*a(floor(n/2)) + n - 2*floor(n/2). - Ralf Stephan, Apr 27 2003 G.f.: (x/(1-x)) * Sum_{k>=0} 3^k*x^2^k/(1+x^2^k). - Ralf Stephan, Apr 27 2003 If the offset were changed to zero, then: a(0) = 0, a(n+1) = f(a(n) + 1, f(a(n)+1) where f(x, y) = if x < 3 and x <> 2 then y else if x mod 3 = 2 then f(y+1, y+1) else f(floor(x/3), y). (End)
We have liminf_{n->infinity} a(n)/n^(log(3)/log(2)) = 1/2 and limsup_{n->infinity} a(n)/n^(log(3)/log(2)) = 1. - Gheorghe Coserea, Sep 13 2015 a(2^k+m) = a(m) + 3^k with 1 <= m <= 2^k and 1 <= k, a(1)=0, a(2)=1. - Paul Weisenhorn, Mar 22 2020 Sum_{n>=2} 1/a(n) = 2.682853110966175430853916904584699374821677091415714815171756609672281184705... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022 a(n) ≍ n^k, where k = log 3/log 2 = 1.5849625007. (I believe the constant varies from 1/2 to 1.) - Charles R Greathouse IV, Mar 29 2024
EXAMPLE
a(6) = 12 because 6 = 0*2^0 + 1*2^1 + 1*2^2 = 2+4 and 12 = 0*3^0 + 1*3^1 + 1*3^2 = 3 + 9.
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
0
1
3, 4
9, 10, 12, 13
27, 28, 30, 31, 36, 37, 39, 40
81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121
MAPLE
t := (j, n) -> add(binomial(n, k)^j, k=0..n):
for i from 1 to 400 do
if(t(4, i) mod 3 <>0) then print(i) fi
# alternative Maple program:
a:= proc(n) option remember: local k, m:
if n=1 then 0 elif n=2 then 1 elif n>2 then k:=floor(log[2](n-1)): m:=n-2^k: procname(m)+3^k: fi: end proc:
# third Maple program:
a:= n-> `if`(n=1, 0, irem(n-1, 2, 'q')+3*a(q+1)):
MATHEMATICA
Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]
Select[Range[0, 400], DigitCount[#, 3, 2] == 0 &] (* Harvey P. Dale, Jan 04 2012 *) Join[{0}, Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 57}]]] (* IWABUCHI Yu(u)ki, Aug 01 2012 *) FromDigits[#, 3]&/@Tuples[{0, 1}, 7] (* Harvey P. Dale, May 10 2019 *)
PROG
(Haskell)
a005836 n = a005836_list !! (n-1)
a005836_list = filter ((== 1) . a039966) [0..]
(Python)
return int(format(n-1, 'b'), 3) # Chai Wah Wu, Jan 04 2015 (Julia)
function a(n)
m, r, b = n, 0, 1
while m > 0
m, q = divrem(m, 2)
r += b * q
b *= 3
end
r end; [a(n) for n in 0:57] |> println # Peter Luschny, Jan 03 2021
CROSSREFS
Cf. A039966 (characteristic function). Cf. A002426, A004793, A005823, A007088, A007089, A032924, A033042- A033052, A054591, A055246, A062548, A065361, A074940, A081601, A081603, A081611, A083096, A089118, A121153, A170943, A185256. For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674. Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
EXTENSIONS
Edited by the Associate Editors of the OEIS, Apr 07 2009
Szekeres's sequence: a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, and thereafter a(n) is smallest number k which avoids any 3-term arithmetic progression in a(1), a(2), ..., a(n-1), k.
(Formerly M0975)
+10
73
1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 82, 83, 85, 86, 91, 92, 94, 95, 109, 110, 112, 113, 118, 119, 121, 122, 244, 245, 247, 248, 253, 254, 256, 257, 271, 272, 274, 275, 280, 281, 283, 284, 325, 326, 328, 329, 334, 335, 337, 338, 352, 353
COMMENTS
That is, there are no three elements A, B and C such that B - A = C - B.
Subtracting 1 from each term gives A005836 (ternary representation contains no 2's). - N. J. A. Sloane, Dec 01 2019 Difference sequence related to Gray code bit sequence ( A001511). The difference patterns follows a similar repeating pattern (ABACABADABACABAE...), but each new value is the sum of the previous values, rather than simply 1 more than the maximum of the previous values. - Hal Burch (hburch(AT)cs.cmu.edu), Jan 12 2004 Sums of distinct powers of 3, translated by 1.
Also, Stanley sequence S(1): see OEIS Index under Stanley sequences (link below). - M. F. Hasler, Jan 18 2016 Named after the Hungarian-Australian mathematician George Szekeres (1911-2005). - Amiram Eldar, May 07 2021 If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+3^n). - Arie Bos, Jul 24 2022
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 164.
Richard K. Guy, Unsolved Problems in Number Theory, E10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(2*k + 2) = a(2*k + 1) + 1, a(2^k + 1) = 2*a(2^k).
a(n) = b(n+1) with b(0) = 1, b(2*n) = 3*b(n)-2, b(2*n+1) = 3*b(n)-1. - Ralf Stephan, Aug 23 2003 G.f.: x/(1-x)^2 + x * Sum_{k>=1} 3^(k-1)*x^(2^k)/((1-x^(2^k))*(1-x)). - Ralf Stephan, Sep 10 2003, corrected by Robert Israel, May 25 2011
EXAMPLE
G.f. = x + 2*x^2 + 4*x^3 + 5*x^4 + 10*x^5 + 11*x^6 + 13*x^7 + 14*x^8 + 28*x^9 + ...
MAPLE
a:= proc(n) local m, r, b; m, r, b:= n-1, 1, 1;
while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*3 od; r
end:
MATHEMATICA
Take[ Sort[ Plus @@@ Subsets[ Table[3^n, {n, 0, 6}]]] + 1, 58] (* Robert G. Wilson v, Oct 23 2004 *) a[1] = 0; h = 180;
Table[a[3 k - 2] = a[k], {k, 1, h}];
Table[a[3 k - 1] = a[k], {k, 1, h}];
Table[a[3 k] = 1, {k, 1, h}];
Table[a[n], {n, 1, h}] (* A189820 *) Flatten[Position[%, 0]] (* A003278 *) Flatten[Position[%%, 1]] (* A189822 *) Table[FromDigits[IntegerDigits[n, 2], 3] + 1, {n, 0, 57}] (* Amit Munje, Jun 03 2018 *)
PROG
(Perl) $nxt = 1; @list = (); for ($cnt = 0; $cnt < 1500; $cnt++) { while (exists $legal{$nxt}) { $nxt++; } print "$nxt "; last if ($nxt >= 1000000); for ($i = 0; $i <= $#list; $i++) { $t = 2*$nxt - $list[$i]; $legal{$t} = -1; } $cnt++; push @list, $nxt; $nxt++; } # Hal Burch
(PARI) a(n)=1+sum(i=1, n-1, (1+3^valuation(i, 2))/2) \\ Ralf Stephan, Jan 21 2014 (Python)
return int(format(n-1, 'b'), 3)+1 # Chai Wah Wu, Jan 04 2015 (Julia)
function a(n)
return 1 + parse(Int, bitstring(n-1), base=3)
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
Similar formula:
If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+4^n) produces A098871; If A_n=(a(1),a(2),...,a(2^n)), then A_(n+1)=(A_n,A_n+2*3^n) produces A191106.
"9ish numbers": decimal representation contains at least one nine.
+10
56
9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298
COMMENTS
The 9ish numbers are closed under lunar multiplication. The lunar primes ( A087097) are a subset. Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018 A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023
LINKS
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
EXAMPLE
E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.
MAPLE
seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020
MATHEMATICA
Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *) Select[Range[300], DigitCount[#, 10, 9]>0&] (* Harvey P. Dale, Mar 04 2023 *)
PROG
(Haskell)
a011539 n = a011539_list !! (n-1)
(GAP) Filtered([1..300], n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019 (Python)
def ok(n): return '9' in str(n)
CROSSREFS
Cf. A088924 (number of n-digit terms).
Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 7.
+10
43
0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85
COMMENTS
Also the set of numbers with no "6" in their base-7 representation; see Gerver-Ramsey, also comments in A020654. - Nathaniel Johnston, Jun 27 2011 Up to the offset, identical to A037470. There are lexicographically earlier, but non-monotonic sequences which do not contain a 7-term AP, e.g., starting with 0,0,0,0,0,0,1,0,... - M. F. Hasler, Oct 05 2014
MAPLE
seq(`if`(numboccur(6, convert(n, base, 7))=0, n, NULL), n=0..85); # Nathaniel Johnston, Jun 27 2011
PROG
(PARI) a(n)=vector(#n=digits(n-1, 6), i, 7^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 7.
+10
29
1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 99
MAPLE
Noap:= proc(N, m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A, forbid, n, c, ds, j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
A[n]:= c;
ds:= convert(map(t -> c-t, A[m-2..n-1]), set);
for j from m-2 to 2 by -1 do
ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]), set);
if ds = {} then break fi;
od;
forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
convert(A, list)
end proc:
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 10.
+10
29
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 49, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 81, 82, 83, 84, 85, 88, 89, 96, 97
MAPLE
Noap:= proc(N, m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A, forbid, n, c, ds, j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
A[n]:= c;
ds:= convert(map(t -> c-t, A[m-2..n-1]), set);
for j from m-2 to 2 by -1 do
ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]), set);
if ds = {} then break fi;
od;
forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
convert(A, list)
end proc:
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 6.
(Formerly M0516)
+10
28
1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 47, 49, 50, 51, 52, 59, 60, 62, 63, 64, 65, 66, 68, 69, 71, 73, 77, 85, 87, 88, 89, 90, 91, 93, 96, 97, 98, 99, 100, 103, 104, 107, 111, 114, 115, 117, 118, 120
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
N:= 100: # to get a(1)..a(N)
A:= Vector(N):
A[1..5]:= <($1..5)>:
forbid:= {6}:
for n from 6 to N do
c:= min({$A[n-1]+1::max(max(forbid)+1, A[n-1]+1)} minus forbid);
A[n]:= c;
ds:= convert(map(t -> c-t, A[4..n-1], set);
if ds = {} then next fi;
ds:= ds intersect convert(map(t -> (c-t)/4, A[1..n-4]), set);
if ds = {} then next fi;
ds:= ds intersect convert(map(t -> (c-t)/3, A[2..n-3]), set);
if ds = {} then next fi;
ds:= ds intersect convert(map(t -> (c-t)/2, A[3..n-2]), set);
forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
PROG
(PARI) A005838(n, show=1, i=1, o=6, u=0)={for(n=1, n, show&&print1(i, ", "); u+=1<<i; while(i++, for(s=1, (i-1)\(o-1), for(j=1, o-1, bittest(u, i-s*j)||next(2)); next(2)); next(2))); i} \\ M. F. Hasler, Jan 03 2016
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
EXTENSIONS
Name and links/references edited by M. F. Hasler, Jan 03 2016
Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 5.
+10
28
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 126, 127, 128, 129, 131, 132, 133
MAPLE
Noap:= proc(N, m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A, forbid, n, c, ds, j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
A[n]:= c;
ds:= convert(map(t -> c-t, A[m-2..n-1]), set);
for j from m-2 to 2 by -1 do
ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]), set);
if ds = {} then break fi;
od;
forbid:= select(`>`, forbid, c) union map(`+`, ds, c);
od:
convert(A, list)
end proc:
MATHEMATICA
t = {1, 2, 3, 4}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Table[Differences[i, 2], {i, s}], {0, 0, 0}], AppendTo[t, n]], {n, 5, 100}]; t (* T. D. Noe, Apr 17 2014 *)
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 6.
+10
28
0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 32, 33, 34, 35, 36, 38, 42, 43, 44, 45, 46, 48, 49, 50, 51, 58, 59, 61, 62, 63, 64, 65, 67, 68, 70, 72, 76, 84, 86, 87, 88, 89, 90, 92, 95, 96, 97, 98, 99, 102, 103, 106, 110, 113, 114, 116, 117, 119, 121
EXAMPLE
5 is excluded since (0,1,2,3,4,5) would be a 6-term AP.
10 is excluded since (0,2,4,6,8,10) would be a 6-term AP.
Idem for 15 and 20.
25 is not excluded, but after 25 (1,6,11,16,21,26) would be a 6-AP, and similarly all of 26 through 32 are excluded.
PROG
(PARI) A020656(n, show=1, i=0, o=6, u=0)={for(n=1, n, show&&print1(i, ", "); u+=1<<i; while(i++, for(s=1, i\(o-1), for(j=1, o-1, bittest(u, i-s*j)||next(2)); next(2)); next(2))); i} \\ M. F. Hasler, Jan 03 2016
CROSSREFS
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
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