cp's OEIS Frontend

The triangle acts as a transform converting any sequence S(k) into a triangle with row sums = S(k). By way of example, begin with S(k), the primes: (2, 3, 5, 7, 11, ...). Add (0, 1, 2, 3, 4, ...) to the sequence getting (prime(n)+(n-1)) = (2, 4, 7, 10, 15, 18, 23, 36, 31, ...) = sequence Q(k). Then replace column 1 (1, 2, 3, ...) of triangle A143349 with sequence Q(k). This = triangle A143350 with row sums prime(n):

2;

4, -1;

7, -1, -1;

10, -2, -1, 0;

...

The A000012 multiplier takes partial sums of A054524 column terms. A051731 is the inverse Mobius transform and A128407 = an infinite lower triangular matrix with mu(n) in the main diagonal and the rest zeros.

Row products are A140700. Similar to table A140256. Can perhaps be seen as taking the moebius function of A126988 times the mangoldt function of A126988.

a(n) ~ log(A003418(n))*n, based on the comment by Hans Havermann in A048272 referring to an argument by Gareth McCaughan.

The exact relation is: lim_{n->Infinity} log(A003418(k))*n = Sum_{x=1..n} Sum_{y=1..k} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0.

Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...

Right border = mu(n), A008683. Row sums = A023900: (1, -1, -2, -1, -4, 2, -6, -1, ...). A054524 = A051731 * A128407.

Left border = d(n), A000010.

Right border = mu(n), A008683.

Row sums = 1.

A128407 * [1,2,3,...] = n*mu(n), A055615: (1, -2, -3, 0, -5, 6, -7, 0, ...). A051731 * A128407 = A054524. A128407 * A051731 = A128408.

Left and right borders = mu(n), A008683. Row sums = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). A128408 * [1,2,3,...] = A063441: (1, -3, -4, 0, -6, 12, ...). A054524 = A051731 * A128407.

When taking matrix powers of the reversed triangle it might be more interesting to start with the first term T(1,1) set to 0.

Three fundamental number theoretic sequences are found from this triangle. The first is the Dirichlet inverse of the Euler totient which are the entries themselves. The Greatest Common Divisor-Fourier transform described by Wolfgang Schramm gives the Möbius function times n =1, -2, -3, 0, ... = A055615, as follows:

1*cos(-2*k*Pi/n) = 1

1*cos(-2*k*Pi/n) -1*cos(-2*k*Pi/n) = -2

1*cos(-2*k*Pi/n) +1*cos(-2*k*Pi/n) -2*cos(-2*k*Pi/n) = -3

The two components in this GCD-Fourier triangle both sum to the sequence 1,0,0,0,0, ... A000007.

1 = 1

1 -1 = 0

1 +1 -2 = 0

...

cos(-2*k*Pi/n) = 1

cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0

cos(-2*k*Pi/n), cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0

...

This latter Fourier transform like triangle is also called the chaotic set by some authors.

The third arithmetic sequence is the von Mangoldt function that can be computed as sums with periods equal to rows in this triangle:

1

log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2))

log(3) = Sum_{n>=0} (1/(n+1) +1/(n+2) -2/(n+3))

log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2) +1/(n+3) -1/(n+4))

log(5) = Sum_{n>=0} (1/(n+1) +1/(n+2) +1/(n+3) +1/(n+4) -4/(n+5))

log(1) = Sum_{n>=0} (1/(n+1) -1/(n+2) -2/(n+3) -1/(n+4) +1/(n+5) +2/(n+6))

...

Also the matrix inverse of the reversal of this number triangle gives the all-ones sequence in the first column. Therefore this number triangle is a companion to A054524.

A subset and also a companion to this triangle in terms of Greatest Common Divisor Fourier transform is A054521, since from A054521 one gets the Mobius function while from this triangle one gets the Möbius function elementwise multiplied by the natural numbers.

The special polynomial found in A199514 is also the solution to the row-wise equations of the symmetric polynomial described in A199514 times the chaotic set or Greatest Common Divisor Fourier transform, so that A199514 is the solution.

Primes have negative signs in a(n). Differs from A055615 at a(30) = 30.

Sum_{k=1..n} T(n,k) = A063524(n).