cp's OEIS Frontend

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)

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.

T(n,k) has the same group structure as A319929, A322630 and A322744. For those arrays, position (3,3) is 5, 7 and 11 respectively. T(3,3) = 13. If we didn't have the formula for these arrays, their entries could be computed knowing one position and applying the arithmetic rules.

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.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

All even terms > 2 appear to be semiprimes of the form 6m+4 (A112774).

The subsequence of odd terms is A307001. - David Lovler, Jan 17 2022

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

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.

There are no more terms up to 5*10^6.

All terms equal 1 (mod 7).

U(n,k) is a commutative and associative array with integer values that depend on whether n and k are odd or even.

U(n,k) = (7*n*k - 5*(n+k-1))/2 when n and k are both odd,

U(n,k) = (7*n*k - 5*n)/2 when n is even and k is odd,

U(n,k) = (7*n*k - 5*k)/2 when n is odd and k is even and

U(n,k) = 7*n*k/2 when n and k are both even.

U(n,1) = n for all n (identity element).

U(n,0) = 0 for all n.

U(n,k) can be expressed as (7*n*k - 5*U(0;n,k))/2, where U(0;n,k) has four cases.

U(0;n,k) = n+k-1 when n and k are both odd,

U(0;n,k) = n when n is even and k is odd,

U(0;n,k) = k when n is odd and k is even and

U(0;n,k) = 0 when n and k are both even.

The ordered list of numbers > 7 that do not appear in array U(n,k) for n and k > 1 can have at most 5 consecutive even numbers and at most 7 consecutive odd numbers. See rows 2 and 3.

U(n,k) is part of a hierarchy of multiplication-like arrays, mentioned in A327263, in which the entries depend on the parity of n and k. U(0;n,k), which is A319929(n,k), is their common parity-dependent component. For i >= 0, U(i;n,k) = (i*n*k - (i-2)*U(0;n,k))/2. In the current sequence U(n,k) = U(7;n,k). Each of these arrays leaves behind a list of numbers that do not appear outside of row 1 and column 1. Think of the prime number sieve.

U(2;n,k) is normal multiplication. For i > 2, these lists are progressively more dense and include even numbers as well as odd numbers. This allows strings of consecutive integers. Here we are interested in the maximum length strings for each i.

The ordered list of numbers > i that do not appear in array U(i;n,k) for n and k > 1 can have at most i-2 consecutive even numbers and at most i consecutive odd numbers. To verify this, look at rows 2 and 3. i consecutive odd numbers cannot interleaf with i-2 even numbers, but i-1 odd numbers can. Thus the longest strings for each i are of length 2i-3. When i is even, m is odd. When i is odd, m is even.

In general, there are more terms when i is odd than when i is even. This is because there are a few ways that i consecutive odd number can overlap i-2 consecutive even numbers.

For this sequence and similar sequences constructed from U(i;n,k), all terms m == 1 (mod i). To prove this, look at gaps of 2i-3 in row 2 of U(i;n,k). The longest strings of consecutive numbers not in U(i;n,k) can occur only for these 2i-3 numbers. The second number before any of the gaps is an even number of the form U(i;2,e) = i*2*e/2 == 0 (mod i) (where e is an even number). The middle of the string, m = U(i;2,e) + i + 1. Thus m == 1 (mod i).

Likewise, for any i, all terms m == 2 (mod i+1).

The following observations are included to expand upon the theme. For odd i in [9..23], the number of terms in [i+2..5*10^6] represented as [i, number of terms] are [9, 3], [11, 8], [13, 3], [15, 0], [17, 3], [19, 3], [21, 0], [23, 3]. For odd i in [25..99], 11 i's have no terms in [i+2..5*10^5], 12 have 1 such term, 9 have 2, 4 have 3, 1 has 4 and 1 has 5 such terms.

The scarcity of terms for even i's is borne out by the observation that up to 5*10^6, for even i in [6..22], the only terms > i+1 occur when i = 14 (1667), i = 20 (3341 and 1663181) and i = 22 (16171). Continuing the observation for even i in [24..140], up to 5*10^5 only 14 i's have a term > i+1.

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.