cp's OEIS Frontend

Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.

A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.

It is conjectured that every positive number appears.

It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.

It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.

[The preceding comments are from Geoffrey Caveney's emails.]

From Michael De Vlieger, Aug 15 2025: (Start)

There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.

There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.

Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

A positive integer k is called a unitary s-Zumkeller number if the set D of proper positive unitary divisors of k can be partitioned as {A,B}, such that the sum of the squares of the elements of A equals the sum of the squares of the elements of B equals (sigma2U(k) - k^2)/2, where sigma2U(k) is the sum of the squares of the unitary divisors of k.

All terms are even numbers.

A term k with a fractional arithmetic mean of divisors is allowed as long as that arithmetic mean evenly divides k+1.

There exist triples such as (19,20,21) and quadruples such as (1,2,3,4) of consecutive numbers where the arithmetic mean of the divisors of every earlier number evenly divides the immediately following number. Are there similar quintuples?

Contains every prime p since (1+p)/2 evenly divides 1+p. - Michael S. Branicky, May 29 2025

Heraclitus (circa 500 BCE) observed that no man can step in the same river twice.

The Heraclitus transform H(S) of a sequence S is formed by starting at 0, and moving s steps to the left or right, where s is any element of S, never visiting any number twice, and moving as close to 0 as possible. In case of a tie, move to the positive term.

The present sequence is the Heraclitus transform of the triangular numbers A000217. For the squares, see A377091. Conjecture: both H(A000217) and H(A000290) contain every (positive or negative) integer. In fact it appears that this property holds whenever S is a monotonically strictly increasing sequence starting with 1. It does not hold for H(A000012), which is A001477.

It is conjectured (see A377090) that every positive integer appears exactly once either here or in A383445.

It is a strong conjecture that every integer appears in A377091, so it is unlikely there will be a second 0 term.

Let G denote the 2-dimensional grid obtained from the square grid Z X Z by deleting the vertices with both coordinates odd and the four edges at each of those vertices (see link). G has vertices with valency either 2 (one coordinate even and one odd, indicated by X) or 4 (both coordinates even, indicated by O). The present sequence is the coordination sequence of G with respect to a vertex of valency 4.

G arises in connection with the six-vertex lattice model of statistical mechanics (see Gorin-Nicoletti).

For n > 1, a(n) = A381116(n) - A381116(n-1) - 1. (This is a trivial consequence of the definitions.)