cp's OEIS Frontend

When r is even, there are r/2 rows for which q - p = r/2. The definition reduces to "numbers congruent to p mod r/2" for these rows. They are kept here for the sake of completeness.

The coefficients in the a(k) formula match those in the e.g.f. If we start with the a(k) formula we get the constant of the e.g.f. by setting a(0)=0.

The sum of two rows with the same r is a row. In fact (using nonnegative integers), any linear combination of rows with the same r yields a row. The same linear combination applied to p, q and r for the rows gives P, Q and R for the combined row. The linear combination also links the steps for the rows with the steps for the combined row.

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.

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

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

When [w,x,y,z] is a row, f(a,b,c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z is associative in the following sense. f((a,b,c),d,e) = f(a,f(b,c,d),e) = f(a,b,f(c,d,e)) for all a,b,c,d,e. f(a,b,c) is commutative because of its symmetry.

For each quadruple, the corresponding f(a,b,c) has a unique zero element (call it theta), meaning f(a,b,theta) = f(a,theta,b) = f(theta,a,b) = theta for all a,b. Theta = -y/x = - x/w. f(a,b,c) also has not one but two identity elements (id_1 and id_2), meaning f(a,id_1,id_1) = f(id_1,a,id_1) = f(id_1,id_1,a) = a for all a and f(a,id_2,id_2) = f(id_2,a,id_2) = f(id_2,id_2,a) = a for all a. Id = (-y +- sqrt(y))/x = theta +- sqrt(y)/x. Thus theta = (id_1 + id_2)/2.

The identity elements and theta are integers when y is a square and x divides sqrt(y).

It is the same to sort by y then by w also to sort by y then by z descending.

When [w,x,y,z] is a row, f(a,b,c) = w*a*b*c + x*(a*b + a*c + b*c) + y*(a+b+c) + z is associative in the following sense. f((a,b,c),d,e) = f(a,f(b,c,d),e) = f(a,b,f(c,d,e)) for all a,b,c,d,e. f(a,b,c) is commutative because of its symmetry.

For each quadruple, the corresponding f(a,b,c) has a unique zero element (call it theta), meaning f(a,b,theta) = f(a,theta,b) = f(theta,a,b) = theta for all a,b. Theta = -y/x = - x/w. f(a,b,c) also has not one but two identity elements (id_1 and id_2), meaning f(a,id_1,id_1) = f(id_1,a,id_1) = f(id_1,id_1,a) = a for all a and f(a,id_2,id_2) = f(id_2,a,id_2) = f(id_2,id_2,a) = a for all a. Id = (-y +- sqrt(y))/x = theta +- sqrt(y)/x. Thus theta = (id_1 + id_2)/2.

The identity elements are integers when y is a square and x divides sqrt(y).

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.

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.

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) = (5*n*k - 3*(n+k-1))/2 when n and k are both odd.

U(n,k) = (5*n*k - 3*n)/2 when n is even and k is odd.

U(n,k) = (5*n*k - 3*k)/2 when n is odd and k is even.

U(n,k) = 5*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.

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

The terms all end in 6 because row 2 of U(n,k) has all numbers that end in 0 or 2 and there are at most 3 consecutive even numbers in the set of numbers not in array U(n,k) excluding the first row and column (see comment for A327263).

There are 119 terms up to 5*10^6.

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).