cp's OEIS Frontend

A digit-wise analog of A049581. Referred to as "box" operation by Eric Angelini.

The binary operator T: N x N -> N is commutative, so this table is symmetric: it does not matter in which direction the antidiagonals are read, and it would be sufficient to specify only the lower half of the square table: see A330238 for this triangle. Zero is the neutral element: T(x,0) = x for all x. Any x is its own inverse or opposite x', as shown by the zero diagonal T(x,x) = 0.

A measure of non-associativity is the "commutator" ((x T y) T x') T y' = ((x T y) T x) T y which would be zero in the associative case, given that x = x' for all x. Here it turns out to be given by 2*A053616, read as a triangle, and rows extended quasi-periodically with period 10, see example.

a(n) = 0 for n in {0, 1, 3, 8, 20, 34} = A224421.

The Fibonacci numbers correspond to sequence A000045.

This sequence is analogous to:

- A051699 (distance to nearest prime),

- A053188 (distance to nearest square),

- A053646 (distance to nearest power of 2),

- A053615 (distance to nearest oblong number),

- A053616 (distance to nearest triangular number),

- A061670 (distance to nearest power),

- A074989 (distance to nearest cube),

- A081134 (distance to nearest power of 3),

The local maxima of the sequence correspond to positive terms of A004695.

a(n) = 0 iff n = A000045(k) for some k >= 0.

a(n) = 1 iff n = A061489(k) for some k > 4.

For any n >= 0, abs(a(n+1) - a(n)) <= 1.

For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).

a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

The length of row n is 1 for n = 1, 2 for n = 2, and 2*n for n >= 3.

The modified modular equivalence relation Modd n is defined, for integer k and positive integer n, by k (Modd n) = k (mod n) if floor(k/n) is even, and -k (mod n) if floor(k/n) is odd. The smallest nonnegative complete residue system modulo n, namely RS(n) = {0, 1, ..., n-1}, is used. See the W. Lang link, Definition 4, eq. (69), p. 25 - 26.

In order to have row length 2*n for all n >= 1 one could use for n = 1 and 2 the imprimitive periods 0, 0 and 0, 1, 0, 1, respectively.

The name Modd n derives from the fact that the multiplicative (but not additive ) group Modd n has the smallest positive reduced residue system with only odd numbers, named RRSodd(n), as elements (for n = 0 RRS(n) = {0}, but here it is taken as {1}). This group is isomorphic to the Galois group G(rho(n)) = Gal(Q(rho(n))/Q), with rho(n) = 2*cos(pi/n). See the W. Lang link.