cp's OEIS Frontend

a(n) is the minimum j such that binomial(j,k) is divisible by n for some k in 0..j.

a(n) is at most equal to A058084(n), the least m such that binomial(m,k) = n for some k.

Sums of triangle rows of the rule 30 automaton, in which the execution of the rule has been modified as such:

In the rule 30 CA, local cell neighborhoods are converted into a number 0-7, which is then used to determine if the central cell will be active in the next generation of the CA.

For this version, when determining this local neighborhood number, any positive nonzero value is treated as a Boolean TRUE. 0 values operate the same way as the conventional version of rule 30.

For example: [0],[0],[14] becomes [False],[False],[True] for the purposes of determining the local state.

Second, instead of writing a 1 when instructed by the 8 sub-rules, instead write the sum of the three local cells.

First take the sum of the values of the left neighbor, the cell itself, and the right neighbor. The next value of the cell will equal this sum if the value of the left neighbor is zero, or if the cell and its right neighbor are both zero.

In other cases the next value of the cell will be zero.

The pattern of nonzero cells is the same as the rule 30 CA (A070950), but the cell values grow as the CA progresses.

It's helpful to define A072272 first: Consider only the four nearest (N,S,E,W) neighbors of a cell together with the cell itself. In the next state, the state of a cell will change if an odd number of these five cells is ON. This is equivalent to taking the sum of all 4 neighbor cells, together with the cell itself, mod 2. A072272 is the total number of active cells per generation. This sequence differs from A072272 because the state of the next cell can be any odd number instead of 1. Instead of applying mod 2 to the sum of the 5 local cells, this variation takes the sum of the 5 local cells, and then sets the value to zero if that value is even.

The generations of A072272 are congruent mod 2 to the generations of this automaton. A consequence of this is that the pattern of active cells is the same between both implementations of the CA. However, the values of the cells in this automaton differ from those in A072272. In this variation the values of cell are not constrained to the range [0,1]. The terms in this sequence are the aggregate values of all cells for each generation, instead of the number of cells.

The automata that generates this sequence operates on a grid of cells c(i,j). The cells in the automata have five possible values, [0-4]. The next generation in the CA is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 5.

Start with a single cell with a value of 1, with all other cells set to 0. For each generation, the term in this sequence c(n) is the aggregate values of all cells in the grid for each discrete generation of the automaton (i.e., not cumulative over multiple generations).

The cellular automaton that generates this sequence has been empirically observed to repeat the number of active cells (4 in this case) if the iteration number N is a power of the modulus + 1. The modulus in this case is 5.

This has been observed to occur with any prime mod and any starting pattern of cells. I'm picking this particular implementation because it's the same as the one used in A102376.

Counting the active (nonzero) cells instead of taking the sum also creates a different but related sequence. This sequence is the sum of each iteration, and cells in this automaton have values 0, 1, 2, 3 or 4. Only for mod 2 are both the sum and active cell counts the same.

The automaton that generates this sequence operates on a grid of cells c(i,j). The cells have three possible values, 0, 1, and 2. The next generation in the CA is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 3.

Start with a single cell with a value of 1, with all other cells set to 0. For each generation, the term in this sequence c(n) is the aggregate values of all cells in the grid for each discrete generation of the automaton (i.e., not cumulative over multiple generations).

The cellular automaton that generates this sequence has been empirically observed to repeat the number of active cells (4 in this case) if the iteration number N is a power of the modulus + 1. The modulus in this case is 3.

This has been observed to occur with any prime modulus and any starting pattern of cells. I'm picking this particular implementation because it's the same as the one used in A102376.

Counting the active (nonzero) cells instead of taking the sum also creates a different but related sequence. This sequence is the sum of each iteration, and cells in this automaton have values 0, 1, or 2. Only for mod 2 are both the sum and active cell counts the same.