A005836 -id:A005836 - cp's OEIS frontend

A262057 Array based on the Stanley sequence S(0), A005836, by antidiagonals.

0, 2, 1, 7, 5, 3, 21, 8, 6, 4, 23, 22, 16, 11, 9, 64, 26, 24, 17, 14, 10, 69, 65, 50, 25, 19, 15, 12, 71, 70, 67, 53, 48, 20, 18, 13, 193, 80, 78, 68, 59, 49, 34, 29, 27, 207, 194, 152, 79, 73, 62, 51, 35, 32, 28, 209, 208, 196, 161, 150, 74, 63, 52, 43, 33, 30

Offset: 1

Max Barrentine, Nov 29 2015

This array is similar to a dispersion in that the first column is the minimal nonnegative sequence that contains no 3-term arithmetic progression, and each next column is the minimal sequence consisting of the numbers rejected from the previous column that contains no 3-term arithmetic progression.
A100480(n) describes which column n is sorted into.
The columns of the array form the greedy partition of the nonnegative integers into sequences that contain no 3-term arithmetic progression. - Robert Israel, Feb 03 2016

			From the top-left corner, this array starts:
   0   2   7  21  23  64
   1   5   8  22  26  65
   3   6  16  24  50  67
   4  11  17  25  53  68
   9  14  19  48  59  73
  10  15  20  49  62  74

Max Barrentine and Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened; n=1..77 from Max Barrentine)

First column is A005836.
First row is A265316.
Cf. A006997, A074940, A100480.

function  A = A262057( M, N )
% to get first M antidiagonals using x up to N
B = cell(1,M);
F = zeros(M,N+1);
countdowns = [M:-1:1];
for x=0:N
    if max(countdowns) == 0
        break
    end
    for i=1:M
        if F(i,x+1) == 0
            newforb = 2*x - B{i};
            newforb = newforb(newforb <= N & newforb >= 1);
            F(i,newforb+1) = 1;
            B{i}(end+1) = x;
            countdowns(i) = countdowns(i)-1;
            break
        end
    end
end
if max(countdowns) > 0
    [~,jmax] = max(countdowns);
    jmax = jmax(1);
    error ('Need larger N: B{%d} has only %d elements',jmax,numel(B{jmax}));
end
A = zeros(1,M*(M+1)/2);
k = 0;
for n=1:M
    for i=1:n
        k=k+1;
        A(k) = B{n+1-i}(i);
    end
end
end % Robert Israel, Feb 03 2016

M:= 20: # to get the first M antidiagonals
for i from 1 to M do B[i]:= {}: F[i]:= {}: od:
countdowns:= Vector(M,j->M+1-j):
for x from 0 while max(countdowns) > 0 do
  for i from 1 do
     if not member(x, F[i]) then
       F[i]:= F[i] union map(y -> 2*x-y, B[i]);
       B[i]:= B[i] union {x};
       countdowns[i]:= countdowns[i] - 1;
     break
    fi
  od;
od:
seq(seq(B[n+1-i][i], i=1..n),n=1..M); # Robert Israel, Feb 03 2016

A006997(A(n, k)) = k - 1. - Rémy Sigrist, Jan 06 2024

A262096 Triangle read by rows: numbers c from the set of arithmetic triples a < b < c (three numbers in arithmetic progression) where a and b are terms of A005836.

Original entry on oeis.org

2, 6, 5, 8, 7, 5, 18, 17, 15, 14, 20, 19, 17, 16, 11, 24, 23, 21, 20, 15, 14, 26, 25, 23, 22, 17, 16, 14, 54, 53, 51, 50, 45, 44, 42, 41, 56, 55, 53, 52, 47, 46, 44, 43, 29, 60, 59, 57, 56, 51, 50, 48, 47, 33, 32, 62, 61, 59, 58, 53, 52, 50, 49, 35, 34, 32

Offset: 1

Max Barrentine, Sep 10 2015

The first term in each row of the triangle is a term of A005823; these are also the local maxima. From this term until the next row, the first differences are A236313.

			Each term is generated from arithmetic sequences started from pairs of terms from A005836. The order is according to the arithmetic triples 0, 1, a(1)=2; 0, 3, a(2)=6; 1, 3, a(3)=5; 0, 4, a(4)=8; 1, 4, a(5)=7; 3, 4, a(6)=5; ...
As a triangle, sequence starts:
   2;
   6,  5;
   8,  7,  5;
  18, 17, 15, 14;
  20, 19, 17, 16, 11;
  24, 23, 21, 20, 15, 14;
  26, 25, 23, 22, 17, 16, 14;
  54, 53, 51, 50, 45, 44, 42, 41;
  ...

Max Barrentine, Table of n, a(n) for n = 1..2016

Cf. A005823, A005836, A074940, A236313, A262097, A262256.

isok(n) = (n==0) || (vecmax(digits(n, 3)) != 2);
lista(nn) = {oks = select(x->isok(x), vector(nn, n, n-1)); for (n=2, #oks, for (k=1, n-1, print1(2*oks[n]-oks[k], ", ");););} \\ Michel Marcus, Sep 12 2015

Name corrected by Max Barrentine, May 24 2016

A262256 List of arithmetic triples aA005836.

Original entry on oeis.org

0, 1, 2, 0, 3, 6, 1, 3, 5, 0, 4, 8, 1, 4, 7, 3, 4, 5, 0, 9, 18, 1, 9, 17, 3, 9, 15, 4, 9, 14, 0, 10, 20, 1, 10, 19, 3, 10, 17, 4, 10, 16, 9, 10, 11, 0, 12, 24, 1, 12, 23, 3, 12, 21, 4, 12, 20, 9, 12, 15, 10, 12, 14, 0, 13, 26, 1, 13, 25, 3, 13, 23, 4, 13, 22

Offset: 1

Max Barrentine, Sep 15 2015

The values a(3n) are A262096.

			0, 1, 2;
0, 3, 6;
1, 3, 5;
0, 4, 8;
...

Max Barrentine, Rows = 1..6048, flattened
Max Barrentine, Rows 1 to 2016

Cf. A005836, A074940, A262096.

Name edited by Max Barrentine, May 24 2016

A374362 a(n) is the least term t of A005836 such that n - t also belongs to A005836.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 3, 3, 4, 0, 0, 1, 0, 0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 0, 0, 1, 0, 0, 1, 3, 3, 4, 0, 0, 1, 0, 0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 27, 27, 28, 27, 27, 28, 30, 30, 31, 27, 27, 28, 27, 27, 28, 30, 30, 31, 36, 36, 37, 36

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): in the ternary expansion of n, replace 1's by 0's and 2's by 1's.

			The first terms, in decimal and in ternary, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     1       2          1
   3     0      10          0
   4     0      11          0
   5     1      12          1
   6     3      20         10
   7     3      21         10
   8     4      22         11
   9     0     100          0
  10     0     101          0
  11     1     102          1
  12     0     110          0
  13     0     111          0
  14     1     112          1
  15     3     120         10

Rémy Sigrist, Table of n, a(n) for n = 0..6561

Cf. A005836, A289814, A374361.

a(n) = fromdigits(apply(d -> [0, 0, 1][1+d], digits(n, 3)), 3)

from gmpy2 import digits
def A374362(n): return int(digits(n,3).replace('1','0').replace('2','1'),3) # Chai Wah Wu, Jul 09 2024

a(n) = A374361(n, 0).
a(n) = n - A374363(n).
a(n) >= 0 with equality iff n belongs to A374361.
a(n) = A005836(1 + A289814(n)).

A374363 a(n) is the greatest term t <= n of A005836 such that n - t also belongs to A005836.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 3, 4, 4, 9, 10, 10, 12, 13, 13, 12, 13, 13, 9, 10, 10, 12, 13, 13, 12, 13, 13, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40, 40, 39, 40, 40, 36, 37, 37, 39, 40, 40, 39, 40, 40, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): in the ternary expansion of n, 2's by 1's.

			The first terms, in decimal and in ternary, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1       2          1
   3     3      10         10
   4     4      11         11
   5     4      12         11
   6     3      20         10
   7     4      21         11
   8     4      22         11
   9     9     100        100
  10    10     101        101
  11    10     102        101
  12    12     110        110
  13    13     111        111
  14    13     112        111
  15    12     120        110

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A005836, A120880, A289831, A374361, A374362.

a(n) = fromdigits(apply(d -> [0, 1, 1][1+d], digits(n, 3)), 3)

a(n) = T(n, A120880(k)-1).
a(n) = n - A374362(n).
a(n) <= n with equality iff n belongs to A005836.
a(n) = A005836(1+A289831(n)).

A374560 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = A005836(n+1) + A005836(k+1).

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 4, 4, 4, 9, 5, 6, 5, 9, 10, 10, 7, 7, 10, 10, 12, 11, 12, 8, 12, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 27, 14, 15, 14, 18, 14, 15, 14, 27, 28, 28, 16, 16, 19, 19, 16, 16, 28, 28, 30, 29, 30, 17, 21, 20, 21, 17, 30, 29, 30, 31, 31, 31, 31, 22, 22, 22, 22, 31, 31, 31, 31

Offset: 0

Rémy Sigrist, Jul 20 2024

For any v >= 0, the value v appears A120880(v) times.

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   3   4   9  10  12  13  27  28  30  31  36
    1 |  1   2   4   5  10  11  13  14  28  29  31  32  37
    2 |  3   4   6   7  12  13  15  16  30  31  33  34  39
    3 |  4   5   7   8  13  14  16  17  31  32  34  35  40
    4 |  9  10  12  13  18  19  21  22  36  37  39  40  45
    5 | 10  11  13  14  19  20  22  23  37  38  40  41  46
    6 | 12  13  15  16  21  22  24  25  39  40  42  43  48
    7 | 13  14  16  17  22  23  25  26  40  41  43  44  49
    8 | 27  28  30  31  36  37  39  40  54  55  57  58  63
    9 | 28  29  31  32  37  38  40  41  55  56  58  59  64
   10 | 30  31  33  34  39  40  42  43  57  58  60  61  66
   11 | 31  32  34  35  40  41  43  44  58  59  61  62  67
   12 | 36  37  39  40  45  46  48  49  63  64  66  67  72

Rémy Sigrist, Table of n, a(n) for n = 0..8255 (antidiagonals for n+k = 0..127 flattened)

Cf. A005836, A120880, A374789, A374790, A374791, A374792.

A(n, k) = fromdigits(binary(n), 3) + fromdigits(binary(k), 3)

A(n, k) = A(k, n).

A(2*n, 2*k) = 3*A(n, k).

A082575 Nonnegative numbers in (3A005836) union (3A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].

Original entry on oeis.org

0, 1, 3, 7, 9, 10, 12, 25, 27, 28, 30, 34, 36, 37, 39, 79, 81, 82, 84, 88, 90, 91, 93, 106, 108, 109, 111, 115, 117, 118, 120, 241, 243, 244, 246, 250, 252, 253, 255, 268, 270, 271, 273, 277, 279, 280, 282, 322, 324, 325, 327, 331, 333, 334, 336, 349, 351, 352

Offset: 1

Emeric Deutsch and Bruce E. Sagan, Dec 05 2003

Numbers k such that the Motzkin number A001006(k) == 1 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Cf. A001006, A005836, A089118, A089119.

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[0, 400], Mod[m[#], 3] == 1 &] (* Jean-François Alcover, Jul 10 2013 *)
max = 150; Sort @ Join[Select[3*Range[0, max], DigitCount[#, 3, 2] == 0 &], Select[3*Range[max] - 2, DigitCount[# + 2, 3, 2] == 0 &]] (* Amiram Eldar, Jun 04 2022 *)

Offset changed to 1 (sequence is a list) by L. Edson Jeffery, Nov 27 2015

A177880 Numbers k such that not all exponents in the prime power factorization of k are in A005836.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 128, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 192, 196, 198, 200, 204, 207

Offset: 1

Vladimir Shevelev, Dec 15 2010

nonn

1 and products of distinct numbers of the form P^(3^k), k>=0, are not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005836.

Select[Range[200], AnyTrue[FactorInteger[#][[;; , 2]], DigitCount[#1, 3, 2] > 0 &] &] (* Amiram Eldar, Aug 31 2020 *)

is_A005836 = lambda n: 2 not in n.digits(base=3)
is_A177880 = lambda n: not all(is_A005836(Integer(m)) for p,m in factor(n)) # D. S. McNeil, Dec 16 2010

Let A(x) be counting function of terms not exceeding x. Then for x tends to infinity, A(x)=C*x+o(x^(0.5+eps), where C=1-Prod{i=p^(3^k)with prime p and k>=0}(1-1/(i^2+i+1)).

Extended by D. S. McNeil, Dec 16 2010

A263488 Positive integers n that can be expressed as the quotient of two elements of A005836.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 19, 21, 22, 25, 27, 28, 30, 31, 34, 36, 37, 39, 40, 55, 57, 58, 61, 63, 64, 66, 67, 70, 73, 75, 76, 79, 81, 82, 84, 85, 88, 90, 91, 93, 94, 97, 100, 102, 103, 106, 108, 109, 111, 112, 115, 117, 118, 120, 121, 163, 165, 166, 169, 171, 172, 174, 175, 178, 181, 183, 184, 187, 189, 190, 192, 193, 196

Offset: 1

Jeffrey Shallit, Dec 02 2015

For each n, a proof of the existence or nonexistence of such a representation can be constructed effectively, by building a finite-state transducer that multiplies by n, and then searching for a path in the corresponding directed graph whose inputs and outputs are labeled only with 0's and 1's. This was used to show, for example, that 529, 592, 601, 616, 5368, and 50281 have no such representation.
It is not hard to show that every element of this sequence lies in an interval bounded by (2/3)*3^n and (3/2)*3^n for some n >= 0. However, not all elements of these intervals have a representation.
It is also not hard to see that if the last nonzero digit of n in base 3 is a 2, then n is not an element of the sequence.
n is in the sequence if and only if 3*n is in the sequence. - Robert Israel, Dec 03 2015

			7 is in the sequence because it can be expressed as 28/4, and in base 3 28 is 1001 and 4 is 11.

Jeffrey Shallit and Robert Israel, Table of n, a(n) for n = 1..1000 (n = 1..399 from Jeffrey Shallit)

Cf. A005836.

F:= proc(N)
  option remember;
  uses GraphTheory;
  local L,G,a,k;
  if N mod 3 = 0 then procname(N/3)
  elif N mod 3 = 2 then return false
  fi;
  k:=  ceil(log[3](2*N/3));
  if N < (2/3)*3^k then return false fi;
  for a from 1 to N-1 do
     L[a]:= {3*a,3*a+1}
  od:
  for a from N to 2*N-1 do
     L[a]:= subs(0=3*N,{3*(a-N),3*(a-N)+1});
  od:
  for a from 2*N to 3*N do
     L[a]:= {};
  od:
  L[3*N+1]:= remove(t -> has(convert(t,base,3),2), {$1..3*N-1}):
  G:= Digraph(3*N+1,[seq(L[a],a=1..3*N+1)]);
  try
    ShortestPath(G,3*N+1,3*N);
  catch "no path from": return false;
  end try;
  true
end proc:
select(F, [$1..1000]); # Robert Israel, Dec 03 2015

A265159 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 5 + 9A005836(2^(k - 1)(2 n - 1)), n,k >= 1.

Original entry on oeis.org

5, 32, 14, 86, 95, 41, 113, 257, 284, 122, 248, 338, 770, 851, 365, 275, 743, 1013, 2309, 2552, 1094, 329, 824, 2228, 3038, 6926, 7655, 3281, 356, 986, 2471, 6683, 9113, 20777, 22964, 9842, 734, 1067, 2957, 7412, 20048, 27338, 62330, 68891, 29525

Offset: 1

L. Edson Jeffery, Dec 03 2015

Conjecture 1: The array contains without duplication all possible "block numbers" defined in A265100.

			Array A begins:
.      5    14    41    122    365    1094    3281     9842    29525
.     32    95   284    851   2552    7655   22964    68891   206672
.     86   257   770   2309   6926   20777   62330   186989   560966
.    113   338  1013   3038   9113   27338   82013   246038   738113
.    248   743  2228   6683  20048   60143  180428   541283  1623848
.    275   824  2471   7412  22235   66704  200111   600332  1800995
.    329   986  2957   8870  26609   79826  239477   718430  2155289
.    356  1067  3200   9599  28796   86387  259160   777479  2332436
.    734  2201  6602  19805  59414  178241  534722  1604165  4812494

Cf. A005836, A055246, A265100, A265104, A265161.

(* Array: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; Grid[Table[a265159[n, k], {n, 9}, {k, 9}]]
(* Array antidiagonals flattened: *)
a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265159[n_, k_] := 5 + 9*a005836[2^(k - 1)*(2 n - 1)]; Flatten[Table[a265159[n - k + 1, k], {n, 9}, {k, n}]]

Conjecture 2: A(n,k) = (A055246(n)*3^k + 1)/2, so the array and A265100 are related to Cantor's ternary set.

G.f. for row n (conjectured): f(n,x) = x*(A265100(n)-(A265100(n)+1)*x)/((1-x)*(1-3*x)).

cp's OEIS Frontend

A262057 Array based on the Stanley sequence S(0), A005836, by antidiagonals.

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A262096 Triangle read by rows: numbers c from the set of arithmetic triples a < b < c (three numbers in arithmetic progression) where a and b are terms of A005836.

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A262256 List of arithmetic triples aA005836.

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A374362 a(n) is the least term t of A005836 such that n - t also belongs to A005836.

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A374363 a(n) is the greatest term t <= n of A005836 such that n - t also belongs to A005836.

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A374560 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = A005836(n+1) + A005836(k+1).

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A082575 Nonnegative numbers in (3*A005836) union (3*A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].

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A177880 Numbers k such that not all exponents in the prime power factorization of k are in A005836.

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A082575 Nonnegative numbers in (3A005836) union (3A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].

A265159 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 5 + 9A005836(2^(k - 1)(2 n - 1)), n,k >= 1.