A143239 - OEIS

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A143239

Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".

REFERENCES

Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.

LINKS

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

FORMULA

Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n, otherwise T(n,k) = 0 if k is not a divisor of n.

Equals A126988 * A128407.

EXAMPLE

First few rows of the triangle are:

2, -1;

3, 0, -1;

4, -2, 0, 0;

5, 0, 0, 0, -1;

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

7, 0, 0, 0, 0, 0, -1;

8, -4, 0, 0, 0, 0, 0, 0;

9, 0, -3, 0, 0, 0, 0, 0, 0;

10, -5, 0, 0, -2, 0, 0, 0, 0, 1;

11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;

12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0;

13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1;

14, -7, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1;

...

Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).

MATHEMATICA

A143239[n_, k_]:= If[Mod[n, k]==0, MoebiusMu[k]*(n/k), 0];

Table[A143239[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

PROG

(Magma)

A143239:= func< n, k | (n mod k) eq 0 select MoebiusMu(k)*(n/k) else 0 >;

[A143239(n, k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024

(SageMath)

def A143239(n, k): return moebius(k)*(n//k) if (n%k)==0 else 0

flatten([[A143239(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Sep 12 2024

CROSSREFS

Cf. A000010 (row sums), A008683, A126988, A128407.