[seqfan] Re: A046790, A046791 obscure, need clarification

[Vladimir Shevelev]

Dear Seqfans,

The first definition in my previous
message, should be corrected for
q>=5 as the following:

"Positive numbers n such that
if n divisible by q, then it divisible by q^3 or the square
of some  other prime, otherwise it divisible by k^2 such
that there is a k_1^2<k^2 and k_1^2==k^2 (mod q)."

In such form it is equivalent to 
"Numbers n such that there is a smaller positive number
 j == n (mod q) such that sqrt(j*n) is integer."
We consider only the case when n is not multiple of q.
Let k^2|n. Set s=(n/k^2)*k_1^2<n.
Then n-s=(n/k^2)*(k^2-k_1^2)==0 (mod q).
Besides, n*s = (n*k_1/k)^2. 

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 13 June 2016 11:56
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A046790, A046791 obscure, need clarification

Dear SeqFan,

Recently we discussed A046790: "Positive numbers
divisible by 8 or by the square of an odd prime" having
many interesting equivalent definitions.
Here a special role plays prime 2.
In case, when a special role plays prime q, one can
consider the sequence: "Positive numbers n such that
if n divisible by q, then it divisible by q^3 or the square
of some  other prime, otherwise it divisible by k^2>=4,
such that k^2==1 (mod q)"
In case q=2 or 3 the "othewise" is equivalent to the
divisibility by the square of some other prime
and, therefore, the cases of divisibility and non-
divisibility of n by q are not separately considered.
There is two equivalent definitions:
a) Numbers n such that there is a smaller positive number
 j == n (mod q) such that sqrt(j*n) is integer.
b) Positive numbers n having a divisor k^2>=4 such that
 n == n/k^2 (mod q).
Thus we have 3 equivalent definitions, while 6 ones in case
q=2.  How to transform others?

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Omar E. Pol [info at polprimos.com]
Sent: 12 June 2016 21:59
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: A046790, A046791 obscure, need clarification

Hi all,

A046790 is also the complement of A078779.

Best regards

Omar


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