[seqfan] Re: two dumb questions
David Seal
david.j.seal at gwynmop.com
Sun Jul 23 11:58:24 CEST 2017
I think there is a typo in the A143104 formula: it says that:
a(i,j) = 1 if j=1 or j|i; 0 otherwise.
But https://en.wikipedia.org/wiki/Redheffer_matrix says:
whose entries a(i,j) are 1 if i divides j or if j = 1; otherwise, a(i,j) = 0.
I.e. there's a 'j divides i' vs 'i divides j' difference - and the Wikipedia definition looks more sensible because the alternative 'j=1' condition is completely redundant in the A143104 definition - j always divides i when j=1.
If that's right, the differences between A051731 and A143104 stem from the fact that T(n,k) as defined in A051731 is omitted from the sequence when k > n (since it is always zero in that case, making A051731 a triangle read by rows:
1
1 - 1
1 - 0 - 1
1 - 1 - 0 - 1
1 - 0 - 0 - 0 - 1
1 - 1 - 1 - 0 - 0 - 1
: : : : : : .
while A143104 is the transpose of that triangle with the omitted zeros included, except that the first column is replaced with ones, and read by upwards antidiagonals:
1 1 1 1 1 1
/ / / / /
1 1 0 1 0 1
/ / / /
1 0 1 0 0 1
/ / /
1 0 0 1 0 0
/ /
1 0 0 0 1 0
/
1 0 0 0 0 1
I should add that as this is the first time I remember ever hearing about Redheffer matrices, I'm not 100% certain about it - so it would be good if anyone more expert can confirm or correct my understanding of what's going on here!
Regards,
David
> On 22 July 2017 at 02:43 Peter Lawrence <peterl95124 at sbcglobal.net> wrote:
>
> A051731: characteristic triangle of divisors of n
>
> A000005: = number of divisors of n = row sums of A051731
>
> 1. How is A051731 different from Redheffer matrix A143104,
> which seems to have same FORMULA definition ?
> a(i,j) = 1 if j=1 or j|i; 0 otherwise.
> ...
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