# a question from a correspodent

Christian Boyer cboyer at club-internet.fr
Fri Oct 26 09:52:10 CEST 2007

Seeing that:

For n = 10 = 2*5,  F(n) = Phi(n) =  4
For n = 14 = 2*7,  F(n) = Phi(n) =  6
For n = 22 = 2*11, F(n) = Phi(n) = 10

Eric Desbiaux is surprised because for any prime p:

F(2p) = Phi(2p)

We can be more precise:

F(2p) = Phi(2p) = p - 1

But not a surprise, because F(n) can be directly computed, without B, C, D,
E:
F(n) is (n/2) - 1 for even n, and is n - 2 for odd n.

We can also add some other similar and obvious remarks, for example:

F(p) = Phi(p) - 1 = p - 2

Christian.

-----Message d'origine-----
De : Max Alekseyev [mailto:maxale at gmail.com]
Envoyé : vendredi 26 octobre 2007 02:38
À : njas at research.att.com; eric.desbiaux at orange.fr
Cc : seqfan at ext.jussieu.fr
Objet : Re: a question from a correspodent

>From what I could get out of English translation of that page, Eric
defines the following sequence (as columns in his Excel file):

A(n) = n
(to list numerical values I will use offset = 5):
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25

B(n) = n^2 - 3*n + 2
12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306,
342, 380, 420, 462, 506, 552

C(n) = [n/2]^2 - [n/2]
[4,] 6, 6, 12, 12, 20, 20, 30, 30, 42, 42, 56, 56, 72, 72, 90, 90,
110, 110, 132, 132

The 5th term 4 (as listed by Eric) does not fit the formula for C(n),
so I enclosed it into brackets.

D(n) = B(n) - C(n) = n^2 - 3*n + 2 - [n/2]^2 + [n/2]
[8,] 14, 24, 30, 44, 52, 70, 80, 102, 114, 140, 154, 184, 200, 234,
252, 290, 310, 352, 374, 420

E(n) = D(n) / 2 = ( n^2 - 3*n + 2 - [n/2]^2 + [n/2] ) / 2
[4,] 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126,
145, 155, 176, 187, 210

F(n) = E(n+1) - E(n) = n-1 for even n and (n-1)/2 for odd n.
[3,] 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12

If F(n) is the sequence in Eric's question, then Yes, it is an integer
sequence. The formula for its terms for n>=6 is given above.

Otherwise, please clarify the question (and the relation to Euler
totient function phi() as I have not got it from that page).

Regards,
Max

On 10/25/07, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Dear Seqfans,  This message just came in.  Maybe someone
> could help?  Some knowledge of French is called for.
> Neil
>
> From: Eric Desbiaux <eric.desbiaux at orange.fr>
> Reply-To: Eric Desbiaux <eric.desbiaux at orange.fr>
> To: njas at research.att.com
> Subject: Question sur fonction phi
>
> Hello Mister
>
> can you help me for this,
> is it an integer sequence?
> thanks
> Best Regards
> Eric
>
>