A265316 -id:A265316 - 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

A268811 Sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a geometric progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 2, 3, 3, 5, 5, 6, 5, 5, 6, 6, 7, 7, 6, 7, 7, 8, 8, 10, 6, 7, 7, 6, 7, 7, 8, 8, 10, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2

Offset: 1

Aaron David Fairbanks, Feb 13 2016

Apparently: all terms belong to A000452, and for any k > 0, the value A000452(k) first appears at index A265316(k+1). - Rémy Sigrist, May 13 2021

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A268811

Cf. A000452, A229037, A265316.

C
```
// See Links section.
```

A268811_list = []
for n in range(1000):
    i, j, b = 1, 1, set()
    while n-2*i >= 0:
        b.add(A268811_list[n-i]**2/A268811_list[n-2*i])
        i += 1
        while j in b:
            b.remove(j)
            j += 1
    A268811_list.append(j)

A368808 Indices of records in A368798.

Original entry on oeis.org

0, 1, 3, 5, 11, 29, 31, 51, 53, 104, 112, 126, 266, 295, 337, 361, 403, 449, 485, 761, 891, 967, 1095, 1175, 1199, 1347, 1480, 1657, 1955, 2229, 2358, 2409, 2537, 3007, 3253, 3497, 3617, 3799, 3926, 3959, 4352, 5038, 5313, 5615, 5847, 6434, 6525, 6972, 7267

Offset: 1

Rémy Sigrist, Jan 06 2024

nonn

Rémy Sigrist, C++ program

Cf. A265316, A368798.

A368798(a(n)) = n - 1.

A348636 Greedy Cantor's Dust Partition.

Original entry on oeis.org

1, 3, 8, 22, 24, 65, 70, 72, 194, 208, 210, 215, 580, 582, 623, 628, 630, 644, 1738, 1740, 1745, 1867, 1869, 1883, 1888, 1890, 1931, 5212, 5214, 5219, 5233, 5235, 5600, 5605, 5607, 5648, 5662, 5664, 5669, 5791, 5793, 15635, 15640, 15642, 15656, 15697, 15699

Offset: 1

Gordon Hamilton, Oct 28 2021

nonn

Starting at 1, consecutively partition the positive integers into sets s(1), s(2), s(3), ... so that no arithmetic sequence of length 3 exists in a set. When choosing s(k), always choose k as small as possible. a(n) = smallest number in s(n).

			S(1) = Cantor's dust 1,2,4,5,10,11,13,14,28,29,31,32,... (A003278)
S(2) = 3,6,7,12,15,16,19,30,33,34,...
S(3) = 8,9,17,18,20,21,35,36,44,...
S(4) = 22,23,25,26,49,50,52,53,...
S(5) = 24,27,51,54,60,63,64,67,...
S(6) = 65,66,68,69,...
S(7) = 70,71,...
S(8) = 72,...
a(1) = min [S(1)] = 1
a(2) = min [S(2)] = 3
a(3) = min [S(3)] = 8
a(4) = min [S(4)] = 22
a(5) = min [S(5)] = 24
a(6) = min [S(6)] = 65
a(7) = min [S(7)] = 70
a(8) = min [S(8)] = 72

MathPickle, Hare vs. Hare, 2017.

Cf. A003278, A005836.

One more than A265316, which is the first row of A262057.

a(n) = A265316(n) + 1.

More terms from David A. Corneth, Nov 03 2021 (computed from A265316).

cp's OEIS Frontend

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

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A268811 Sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a geometric progression.

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A368808 Indices of records in A368798.

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A348636 Greedy Cantor's Dust Partition.

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