A127649 A127648 * A054523 as infinite lower triangular matrices.

1, 2, 2, 6, 0, 3, 8, 4, 0, 4, 20, 0, 0, 0, 5, 12, 12, 6, 0, 0, 6, 42, 0, 0, 0, 0, 0, 7, 32, 16, 0, 8, 0, 0, 0, 8, 54, 0, 18, 0, 0, 0, 0, 0, 9, 40, 40, 0, 0, 10, 0, 0, 0, 0, 10, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 48, 24, 24, 24, 0, 12, 0, 0, 0, 0, 0, 12, 156, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 84

Offset

1,2

Comments

Natural number transform of A054523.

Row sums = n^2, left column = A002618

Links

Table of n, a(n) for n=1..92.

Formula

T(n,k)=n*A054523(n,k). - R. J. Mathar, Nov 01 2007

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x - y. - Mats Granvik, Oct 08 2023

Example

First few rows of the triangle are:
1;
2, 2;
6, 0, 3;
8, 4, 0, 4;
20, 0, 0, 0, 5;
12, 12, 6, 0, 0, 6;
42, 0, 0, 0, 0, 0, 7;
...

Maple

A054523 := proc(n, k) if n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi ; end: A127649 := proc(n, k) A054523(n, k)*n ; end: for n from 1 to 20 do for k from 1 to n do printf("%d, ", A127649(n, k)) ; od: od: # R. J. Mathar, Nov 01 2007

Crossrefs

Cf. A127648, A002618, A054523.

Sequence in context: A163119 A091085 A011144 * A274440 A378208 A199220

Adjacent sequences: A127646 A127647 A127648 * A127650 A127651 A127652

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jan 22 2007

Extensions

More terms from R. J. Mathar, Nov 01 2007

Status

approved