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Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.

Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

Contains only semiprimes (A001358).

sum{n>=1, k>=1} 1/T(n,k)^s = P(s)^2, where P is the Prime Zeta Function. - Enrique Pérez Herrero, Jun 24 2012

Or, multiplication table of the positive Fibonacci numbers read by antidiagonals.

Or, triangle of products of nonzero Fibonacci numbers.

Or, a two-dimensional square Fibonacci array read by antidiagonals, with offset 1: T(1,1) = T(1,2) = T(2,1) = T(2,2) = 1; thereafter T(m,n) = max {T(m,n-2) + T(m,n-1), T(m-2,n) + T(m-1,n), T(m-2,n-2) + T(m-1,n-1)}. If "max" is changed to "min" we get A283845. - N. J. A. Sloane, Mar 31 2017

Row sums are A001629 (Fibonacci numbers convolved with themselves.). The main diagonal and first subdiagonal are Fibonacci numbers, for other entries T(n,k) = T(n-1,k) + T(n-2,k). The central numbers form A006498. - Gerald McGarvey, Jun 02 2005

Alternating row sums = (1, 0, 3, 0, 8, ...), given by Fibonacci(2n) if n even, else zero.

Row n = edge-counting vector for the Fibonacci cube F(n+1) embedded in the natural way in the hypercube Q(n+1). - Emanuele Munarini, Apr 01 2008

The augmentation of A058071 is the triangle A193595. To fit the definition of augmented triangle at A103091, it is helpful to represent A058071 using p(n,k)=F(k+1)*F(n+1-k) for 0<=k<=n. - Clark Kimberling, Jul 31 2011

T(n,k) = number of appearances of a(k) in p(n) in the n-th convergent p(n)/q(n) of the formal infinite continued fraction [a(0), a(1), ...]; e.g., p(3) = a(0)*a(1)*a(2)*a(3) + a(0)*a(1) + a(0)*a(3) + a(2)*a(3) + 1. Also, T(n,k) = number of appearances of a(k+1) in q(n+1); e.g., q(3) = a(1)*a(2)*a(3) + a(1) + a(3). - Clark Kimberling, Dec 21 2015

Each row is a palindrome, and the central term of row 2n is the square of the F(n+1), where F = A000045 (Fibonacci numbers). - Clark Kimberling, Dec 21 2015

Also called Hosoya's triangle, after the Japanese chemist Haruo Hosoya (b. 1936). - Amiram Eldar, Jun 10 2021

Square array A(n,k), n >= 1, k >= 1, read by descending antidiagonals.

The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of GF(2) polynomial rings such as GF(2)[x,y]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. This set has a natural arrangement in a square array, given by A329050(i,j) = prime(i+1)^(2^j), i >= 0, j >= 0. The most meaningful generating set for the additive group of GF(2)[x,y] is {x^i * y^j: i >= 0, j >= 0}, which similarly forms a square array. All this makes A329050(i,j) especially appropriate to be the image (under an isomorphism) of the GF(2) polynomial x^i * y^j.

Using g to denote the intended isomorphism, we specify g(x^i * y^j) = A329050(i,j). This maps minimal generating sets of the additive groups, so the definition of g is completed by specifying g(a+b) = A059897(g(a), g(b)). We then calculate the image under g of polynomial multiplication in GF(2)[x,y], giving us this sequence as the matching multiplicative operator for an isomorphic ring over the positive integers. Using f to denote the inverse of g, A[n,k] = g(f(n) * f(k)).

See the formula section for an alternative definition based on the A329050 array, independent of GF(2)[x,y].

Closely related to A306697 and A297845. If A059897 is replaced in the alternative definition by A059896 (and the definition is supplemented by the derived identity for the absorbing element, shown in the formula section), we get A306697; if A059897 is similarly replaced by A003991 (integer multiplication), we get A297845. This sequence and A306697, considered as multiplicative operators, are carryless arithmetic equivalents of A297845. A306697 uses a method analogous to binary-OR when there would be a multiplicative carry, while this sequence uses a method analogous to binary exclusive-OR. In consequence A(n,k) <> A297845(n,k) exactly when A306697(n,k) <> A297845(n,k). This relationship is not symmetric between the 3 sequences: there are n and k such that A(n,k) = A306697(n,k) <> A297845(n,k). For example A(54,72) = A306697(54,72) = 273375000 <> A297845(54,72) = 22143375000.

(Conjecture) Let N=2*n and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (see [Jeffery]) associated with N. Define the Chebyshev polynomials of the second kind by the recurrence U_0(x)=1, U_1(x)=2*x and U_r(x)=2*x*U_(r-1)(x)-U_(r-2)(x) (r>1). Define the column vectors V_(k-1)=(U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where B^T denotes the transpose of matrix B. Let S_N=[V_0,V_1,...,V_(n-1)] be the n X n matrix formed by taking the components of vector V_(k-1) as the entries in column k-1 (V_(k-1) gives the ordered spectrum of A_{N,k-1}). Let X_N=[S_N]^T*S_N, and let [X_N](i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then also T(n,k)=[X_N](k-1,k-1); that is, row n of the triangle is given by the main diagonal entries of X_N. Hence T(n,k) is the sum of squares T(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]=[V_(k-1)]^T*V_(k-1). - L. Edson Jeffery, Jan 20 2012

Conjecture that antidiagonal sums are A023855. - L. Edson Jeffery, Jan 20 2012

Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A066680 U {1}; this is the only one that contains its own row numbers only once. - Peter Munn, Dec 04 2019

The union of N, 2N, 3N, ..., where N = {1, 2, 3, 4, 5, 6, ...}. In other words, the numbers {m*n, m >= 1, n >= 1} sorted into nondecreasing order.

Considering the maximal rectangle in each of the Ferrers graphs of partitions of n, a(n) is the smallest such maximal rectangle; a(n) is also an inverse of A006218. - Henry Bottomley, Mar 11 2002

The numbers in A003991 arranged in numerical order. - Matthew Vandermast, Feb 28 2003

Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - Michel Lagneau, Jan 04 2012

The number 1 appears only once, primes appear twice, squares of primes appear thrice. All other positive integers appear at least four times. - Alonso del Arte, Nov 24 2013

Associative multiplication-like table whose values depend on whether n and k are odd or even.

Associativity is proved by checking the formula with eight cases of three odd and even arguments. T(n,k) is distributive as long as partitioning an even number into two odd numbers is not allowed.

All the U(i;n,k) mimic the ordinary multiplication table in that they are commutative, associative, have identity element 1 and have 0. However (except when i=2) they are partially distributive, meaning that distributivity works except if an even number is partitioned into a sum of two odd numbers. Only when i=2, the odd-even-dependent A319929 term disappears and normal distributivity holds.

U(0;n,k) = A319929(n,k);

U(1;n,k) = A322630(n,k);

U(2;n,k) = n*k;

U(3;n,k) = A322744(n,k);

U(4;n,k) = A327259(n,k);

U(i;n,k) = 2i*floor(n/2)*floor(k/2) + A319929(n,k).

Row 1 is 2p-1 where p is a prime number (A076274 without 1).

Row 2 is the prime numbers.

Row 3 is A307002.

Row 4 is A327261.

The i-th row of T(n,k) consists of numbers that sieve out of the array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2, in numerical order.

From David Lovler, Sep 02 2020: (Start)

Row 1 has no even numbers. Row 2 has one even number. Generally, the even numbers of the i-th row start with i-1 consecutive even numbers (from 2). This is because U(i;2,2) = 2*i gives the first even number not in row i.

Row 3 seems to have even numbers that, after 2, coincide with A112774 which has an infinite number of terms. For i > 3, as i increases, row i has a denser presence of even numbers, thus each row has an infinite number of even terms.

Generalization of the twin prime conjecture: Since row 2 is the prime numbers, we can observe the twin prime conjecture that after the first three odd primes, the sprinkling of pairs of consecutive prime numbers never ends. Concerning just odd terms, a similar conjecture can be stated for rows i >= 3. Row 3 starts with four odd numbers then the sprinkling of three consecutive odd number never ends. Row 4 starts with five odd numbers then the sprinkling of four consecutive odd numbers never ends. The pattern continues as row i starts with i+1 odd numbers then the sprinkling of i consecutive odd numbers never ends. We can take this back to row 1 which starts with two odd numbers then continues with isolated odd numbers.

Studying the even terms, there is an analog to the above generalization of the twin prime conjecture. Row 3 starts with two even numbers then continues with isolated even numbers. Row 4 starts with three even numbers then the sprinkling of pairs of consecutive even numbers never ends. Row 5 starts with four even numbers then the sprinkling of three consecutive even numbers never ends. The pattern continues as row i starts with i-1 even numbers then the sprinkling of i-2 consecutive even numbers never ends.

(End)

For any m > 0:

- let F(m) be the set of distinct Fermi-Dirac primes (A050376) with product m,

- for any i >=0 0 and j >= 0, let f(prime(i+1)^(2^i)) be the lattice point with coordinates X=i and Y=j (where prime(k) denotes the k-th prime number),

- f establishes a bijection from the Fermi-Dirac primes to the lattice points with nonnegative coordinates,

- let P(m) = { f(p) | p in F(m) },

- P establishes a bijection from the nonnegative integers to the set, say L, of finite sets of lattice points with nonnegative coordinates,

- let Q be the inverse of P,

- for any n > 0 and k > 0:

T(n, k) = Q(P(n) + P(k))

where "+" denotes the Minkowski addition on L.

This sequence has similarities with A297845, and their data sections almost match; T(6, 6) = 30, however A297845(6, 6) = 90.

This sequence has similarities with A067138; here we work on dimension 2, there in dimension 1.

This sequence as a binary operation distributes over A059896, whereas A297845 distributes over multiplication (A003991) and A329329 distributes over A059897. See the comment in A329329 for further description of the relationship between these sequences. - Peter Munn, Dec 19 2019

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

This array defines a binary operation on the nonnegative integers that matches up the zeros in the binary representation of each operand (starting from the right, and including as many leading zeros as necessary) and concatenates the two (possibly null) strings of ones to the right of each matched pair of zeros. See the examples. - Peter Munn, Feb 14 2021.

As such it could be useful for implementing multiplication, say, in Turing machines, with a "tape-like" unary-binary encoding of the prime factorization of n (A156552). However, such representation is not very useful if addition or subtraction is also needed.