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Every positive integer, m, is the product of a unique subset, S(m), of the numbers listed in A050376 (primes, squares of primes etc.) The Fermi-Dirac factors of m are the members of S(m). So T(n,k) is the product of the members of (S(n) U S(k)).

Old name: Table a(i,j) = product prime(k)^(Ei(k) OR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; OR is the bitwise operation on binary representation of the exponents.

Analogous to LCM, with OR replacing MAX.

A003418-analog seems to be A066616. - Antti Karttunen, Apr 12 2017

Considered as a binary operation, the result is the lowest common multiple of the squarefree parts of its operands multiplied by the square of the operation's result when applied to the square roots of the square parts of its operands. - Peter Munn, Mar 02 2022

For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).

This table has many similarities with A248601.

For any n > 0 and m > 0, f(n * m) = f(n) + f(m).

Also, f(1) = 0 and f(2) = 1.

The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.

See A297473 for the main diagonal of T.

As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020

From Peter Munn, Jun 16 2021: (Start)

The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.

Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.

There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.

There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.

(End)

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.

Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.

In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...

In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

An equivalent definition is: a(1) = 1, a(2) = 2; and for n > 2, a(n) = least positive integer > a(n-1) and not of the form a(i)*a(j) for 1 <= i < j < n.

a(2) to a(29) match the initial terms of A000028. [corrected by Peter Munn, Mar 15 2019]

This has a simpler definition than A000028, but the resulting pair lacks the crucial property of the A000028/A000379 pair (see the comment in A000028). - N. J. A. Sloane, Sep 28 2007

Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

From Vladimir Shevelev, Apr 05 2013: (Start)

1) The sequence does not contain (for example) 140, so is different from A000028.

2) Representation of numbers which are absent in the sequence as a product of two different terms of the sequence is, generally speaking, not unique. For example, 210 = 2*105 = 3*70 = 5*42 = 7*30.

(End)

Excluding a(1) = 1, the lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 distinct terms. Removing the latter distinctness requirement, the sequence becomes A026424; and the equivalent sequence where the product is of 2 or more distinct terms is A050376. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - Peter Munn, Mar 15 2019

A multiplicative semigroup; if m and n are in the sequence then so is m*n. - David James Sycamore, Jul 17 2018

From Peter Munn, Jul 19 2020: (Start)

Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.

A number is listed if and only if it has an even number of odd prime factors, counting repetitions; equivalently, if and only if it is the product of a term of A046337 and a power of 2 (term of A000079).

(End)

Analogous to GCD, with AND replacing MIN.

The parity of the sequence is E,D,E,E,E,D,E,E,E,D,E,E,E,D,E,E,E,D,E,E,E,D,..., that is, an D followed by three E's from the second term onwards.

Closely related to A035263: if A035263(n) = 1, a(n) = 2n; otherwise a(n)=n/2. - Franklin T. Adams-Watters, Feb 02 2006

This permutation is self-inverse. This is the case r=2 of sequences where a(n)=floor(n/r) if floor(n/r)>0 and not already in the sequence, a(n) = floor(n*r) otherwise. All such sequences (for r>=1) are permutations of the natural numbers. - Franklin T. Adams-Watters, Feb 06 2006

Take the list of positive integers L. At each step n swap L(n) with L(2*L(n)). - Ali Sada, Jun 18 2025

This sequence is a permutation of A050376, so every positive integer is the product of a unique subset, S_factors, of its terms. If we restrict S_factors to be chosen from a subset, S_0, consisting of numbers from specified rows and/or columns of this array, there are notable sequences among those that may be generated. See the examples. Other notable sequences can be generated if we restrict the intersection of S_factors with specific rows/columns to have even cardinality. In any of the foregoing cases, the numbers in the resulting sequence form a group under the binary operation A059897(.,.).

Shares with array A246278 the property that columns grow downward by iterating A003961, and indeed, this array can be obtained from A246278 by selecting its columns 1, 2, 8, 128, ..., 2^((2^k)-1), for k >= 0.

A(n,k) is the image of the lattice point with coordinates X=n and Y=k under the inverse of the bijection f defined in the first comment of A306697. This geometric relationship can be used to construct an isomorphism from the polynomial ring GF(2)[x,y] to a ring over the positive integers, using methods similar to those for constructing A297845 and A306697. See A329329, the ring's multiplicative operator, for details.

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 139, 1427, 14207, 141970, 1418563, 14183505, 141834204, 1418330298, 14183245181, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.4183... . - Amiram Eldar, Sep 10 2022