cp's OEIS Frontend

For any positive number with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), we build a XOR-triangle with (e_m, ..., e_1) as top row, and having each entry in the subsequent rows be the XOR of the two values above it. This sequence lists integers whose XOR-triangle has 3-fold rotational symmetry. A334990 gives the second row of such XOR-triangles for numbers that are not powers of 2.

This sequence has strong connections with A334556: for any n > 0 and k > 0, A019565(A334556(n))^k belongs to this sequence.

Every power of 2 belongs to the sequence.

If m belongs to this sequence, then m^2 also belongs to this sequence.

The n-th row has A060547(n) terms.

Every positive term of A334556, say m, appears in row A070939(m).

For any nonnegative number n, the EQ-triangle for n is built by taking as first row the binary expansion of n (without leading zeros), having each entry in the subsequent rows be the EQ of the two values above it (a "1" indicates that these two values are equal, a "0" indicates that these values are different).

The second row in such a triangle has binary expansion given by A279645.

If m belongs to this sequence, then A030101(m) also belongs to this sequence.

All positive terms are odd.

This sequence is a variant of A334556; here we use bitwise EQ, there bitwise XOR.

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition b(m) of m and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit. We may plot T(m) as an equilateral triangle, since each iteration decrements the binary integer length L of the output until we have L = 1.

The XOR function used here requires two inputs; if the inputs agree, the output is 0, else 1.

A rotationally-symmetrical XOR-triangle (RST) is one whose appearance is the same when rotated 120 degrees.

A zero triangle of side length k arises when we have a run of (k + 1) 1s in the preceding iteration.

This sequence contains m that produce T(m) with a recordsetting side length of its largest zero-triangle. For 1 < n < 3, T(a(n)) only has eccentric zero triangles. T(a(4)) has a singleton zero at center, thus a central zero triangle (CZT) of k = 1. For n > 4, all T(a(n)) have CZTs.

The number 543 = A281482(4); we observe that A281482(2^i) produces RSTs, and only for 0 <= i <= 2 do we have eccentric zero triangles larger than any possible CZT. For A281482(2^3) = 131583, the side length of its eccentric zero triangles prove much smaller than the largest possible CZT.

Since this sequence wants to maximize the side length k of the largest triangle, we see that the largest triangle possible is the CZT. Let j be the "frame width" or number of iterations required to generate the first run of 0s in the CZT. We note j >= 2, since j = 1 requires a run of (k + 1) ones delimited by at least 1 zero; such a width implies that these zeros occur at the beginning and end of b(m). However, beginning binary notation with a leading zero is not permitted. Therefore, if it is possible, we will see T(m) with j > 1.

The numbers that produce record-setting m are the smaller of the binary reverse of m, therefore binary weight favors the least significant digits. Thus we see a 1 followed by a number of zeros in a "prefix" A that, along with a suffix C, must have the same number of bits.

For RSTs with a CZT, we have only one way to produce a solid run of (k + 1) zeros, that is, by dithering bits, which necessitates paired 0 and 1, therefore, we have even k for n > 4.

A run-length formula for a(n) with n > 4 is 12..i(11)..3, meaning that we have 1 one, 2 zeros, any number i of paired 1-0 bits, and a run of 3 ones. Aside from the reversal of this pattern, which puts a greater binary weight in the most significant 3 bits, there is no other way to construct a smaller (or any) CZT with frame size j = 2.

This equates to linear recurrence kernel (5, -4) starting with {2391, 9559} (though 39 is part of the same trajectory).