cp's OEIS Frontend

Row a(n) of A067255 = row A334556(n) of A333624.

An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n 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.

Let T(n,k) address the terms in the k-th position of row n in A333624.

This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334556(n). To decode a(n) => A333624(A334556(n)), we use A067255(a(n)).

Row a(n) of A067255 = row A334769(n) of A333624.

An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n 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.

Let T(n,k) address the terms in the k-th position of row n in A333624.

This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) => A333624(A334769(n)), we use A067255(a(n)).

All terms of A055204 are squarefree by definition, therefore we can compress the terms of A055204 by interpreting the terms of reverse(A067255(A055204(n))) as a binary number and converted to decimal.

To compute T(n, k):

- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),

- to "multiply" two prime numbers: take the smallest,

- to "add" two prime numbers: take the largest,

- for example, for T(12, 14):

(11 7 5 3 2)

12 --> 1 2

14 --> x 1 0 0 1

---------

1 1

0 0

0 0

+ 1 1

-----------

1 1 0 1 1 --> 462 = T(12, 14)

This sequence is closely related to lunar multiplication (A087062):

- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,

- there is a natural bijection f from S_b to the set of nonnegative integers:

f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,

- let g be the inverse of f,

- then for any numbers n and k in S_b, we have:

T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,

- the corresponding addition table is A003990.

The sequence is also readable as an irregular triangle by rows in which row n lists the terms divisible by 2^k but not by 2^(k+1).

Numbers m in A363063 are products of prime powers p(j)^S(j), j = 1..N, where p(j) is the j-th prime, such that p(j+1)^S(j+1) < p(j)^S(j). As consequence of definition of A363063, S(j) > S(j+1), hence multiplicities S(j) are distinct. Consequently, A363063 is a subset of A025487; m is a product of primorials. A025487 in turn is a subset of A055932.

These qualities enable us to write an algorithm that increments S(j) or drops the last term in S until we can increment S(j) to attain a solution. This algorithm generates terms in lexicographic order as described in the Name. The same qualities enable expression of m = Product p(j)^S(j) instead as Sum 2^(S(j)-1), a strictly increasing sequence.

Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.

Observations regarding prime power decomposition of terms in a(0..20737):

For n > 300, most terms are in A361098 but not in A286708. A303606 is a subset of A286708, which is a subset of A361098, which in turn is a subset of A126706, numbers that are neither prime powers nor squarefree.

a(34) = 36 is the only term in A286708 (more specifically, in A303606).

a(35) = 2 is the last prime term.

a(29) = 8 is the only composite prime power.

a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.

a(191) = 2310 is the last primorial term.

a(1055) = 2207550414530882190 is the last squarefree term. If there are further squarefree terms a(n), n is likely to belong to -1 (mod 24).

a(7055) = 1733187515208453605856007490304335826298500960 is the last term that is not in A361098. a(n) not in A361098 is likely to belong to -1 (mod 24).

In other words, numbers k that set records for d(k) - f(k), where d = A000005 and f = A008479.

Largest primorial in this sequence is A002110(4) = 210.

The primorials A002110(0..4) are the only squarefree numbers in this sequence.

{a(n)} \ A002110(0..4) is contained in A126706.

Not a subset of A060735; a(13) = 2520 is not in A060735. Though common for small n, the set of a(n) in A060735 is likely finite; the restriction is connected with the finite number of primorials in the sequence.

Not a subset of A025487 or A055932; a(19) = 32760 is the smallest term without a primorial kernel.

The prime shape of a(n) appears to feature exponents m of prime power factors p^m | a(n) that are nonincreasing as pi(p) increases.

Let f(k) = A381096(k) = k - phi(k) - tau(k/rad(k)) = k - A000010(k) - A005361(k), where phi = A000010, tau = A000005, and rad = A007947. This sequence contains k such that f(k) > f(j) for j < k as k increases.

Apart from a(1) = 1, terms are in A024619.

Conjecture 1: For i > 1, A002110(i) is in this sequence.

Conjecture 2: Intersection with A001694 (i.e., in A286708) is {900, 1800}.

In the Kac reference this array is called rho_{p}(n) := 1 if p divides n else 0.

The row length sequence is A061395(n),n>=2: [1,2,1,3,2,4,1,2,3,5,2,6,4,3,...] (the index of the largest prime dividing n). All row entries beyond these numbers are 0, hence they are not shown. The n=1 row would have 0 for all entries.

The column sequences (without leading zeros) give for m>=1 periodic sequences with the period: 1 followed by p(m)-1 zeros. They start with n=p(m) := A000040(m).

The rows of this triangle also give all the ordered ways that a finite number of integers can be arranged so that their partial sums, from left to right, are all nonnegative and their total sum is positive.