cp's OEIS Frontend

While array A327259 has many properties of the multiplication table, one way the numbers that sieve out of the array fail to be prime numbers is that unique factorization does not hold. Some numbers have two or more decompositions.

For i >= 2, A327259(i, a(n)) is in the sequence.

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)

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

The subsequence of odd terms is A327260. - David Lovler, Jan 23 2022

Terms > 5 have 7 for their units digit. This is because the units digit of A327261 terms can't be 0 or 3 (row 3 of A327259 has all numbers that end in 0 or 3) and there are at most 2 consecutive even terms (see comment for A327263).

The first bisection is A139098, the second bisection is A102083.

Column k is an arithmetic progression with difference 2*A008794(k).

Odd rows of A133728 triangle are contained in row 0.

For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

Terms are the odd numbers not appearing in array A327259 with its first row and column omitted.

T(n,k) is commutative, associative, has identity element 1 and has 0. Also it is distributive except when an even number is partitioned into two odd numbers. Thus it has a multiplicative structure similar to that of A319929, A322630, A322744 and A327259 except that T(odd,odd) is not always odd, T(even,even) is not always even and T(odd,even) is not always even.

T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334923) i is being incremented by 1/2. When i is incremented by 1/4 or less, array values cease to be all integers, although all of the multiplication rules still hold.

T(n,k) is commutative, associative, has identity element 1 and has 0. Also it is distributive except when an even number is partitioned into two odd numbers. Thus it has a multiplicative structure similar to that of A319929, A322630, A322744 and A327259 except that T(odd,odd) is not always odd, T(even,even) is not always even and T(odd,even) is not always even.

T(n,k) is in the same form as the supplementary arrays of A327263 called U(i;n,k). Here (and in A334922) i is being incremented by 1/2. When i is incremented by 1/4 or less, array values cease to be all integers, although all of the multiplication rules still hold.