A143349 - OEIS

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A143349

Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

1, 2, -1, 3, -1, -1, 4, -2, -1, 0, 5, -2, -1, 0, -1, 6, -3, -2, 0, -1, 1, 7, -3, -2, 0, -1, 1, -1, 8, -4, -2, 0, -1, 1, -1, 0, 9, -4, -3, 0, -1, 1, -1, 0, 0, 10, -5, -3, 0, -2, 1, -1, 0, 0, 1, 11, -5, -3, 0, -2, 1, -1, 0, 0, 1, -1, 12, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, 13, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, -1, 14, -7, -4, 0, -2, 2, -2, 0, 0, 1, -1, 0, -1, 1

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OFFSET

1,2

COMMENTS

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):

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.

LINKS

Table of n, a(n) for n=1..105.

EXAMPLE

First few rows of the triangle:

2, -1;

3, -1, -1;

4, -2, -1, 0;

5, -2, -1, 0, -1;

6, -3, -2, 0, -1, 1;

7, -3, -2, 0, -1, 1, -1;

8, -4, -2, 0, -1, 1, -1, 0;

9, -4, -3, 0, -1, 1, -1, 0, 0;