[seqfan] Re: Composites in the i*j multplication table A333996

rgwv at rgwv.com rgwv at rgwv.com
Sat Oct 3 16:57:16 CEST 2020


A334454: f[n_] := Length[ Select[ Union[ Flatten[ Table[j*k, {j, n}, {k, n}]]], CompositeQ@# &]]; Array[f, 53]
which he submitted three days later using the same title.

A333996 f[n_] := Length[ Select[ Flatten[ Table[j*k, {j, n}, {k, j, n}]], CompositeQ@# &]]; Array[f, 53]

In A333996, the iterator only looks at half the multiplication table.

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Peter Kagey
Sent: Friday, October 2, 2020 12:36 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Composites in the i*j multplication table A333996

This sequence appears to be something like the number of pairs (i,j) with 1 <= i <= j <= n such that the product is composite.

So 4 is counted twice in your set, both as 1*4 and as 2*2. There’s a comment about this in the discussion history, but it didn’t make it to the published sequence. 

So a(4)=7 with {1*4, 2*2, 2*3, 2*4, 3*3, 3*4, 4*4}. 

pk

> On Oct 2, 2020, at 3:51 AM, Richard J. Mathar <mathar at mpia-hd.mpg.de> wrote:
> 
> Does someone understand the a(4)=7 in the A333996(4) count for 
> composites in the 4x4 multiplication table?
> I get a(4)=6, that is the cardinality of {4,6,8,9,12,16}, and a 
> sequence like 0, 1, 3, 6, 10, 14, 20, 25, 31, 37...
> See also A027424 (cardinality of the set of all i*j) and A000720 
> (number of primes in the multiplication table).
> 
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