[seqfan] Re: PF seqence and A151659

Benoît Jubin benoit.jubin at gmail.com
Fri Jul 3 00:06:53 CEST 2009


Thanks for adding these sequences.  By the way, your previous email is
there: http://list.seqfan.eu/pipermail/seqfan/2009-April/001331.html

I would like to stress the fact that definitions are different from
the algorithms used to compute the sequence (although the distinction
is not always clear).  In particular, definitions should be
understandable by people who don't know much about programming or
coding, and in my opinion the use of pseudo-code should be avoided for
definitions.  Of course, effective algorithms are more than welcome,
but rather as comments.
Also, I'm always puzzled when I see the indefinite article a/an used
in a definition.

I hope that I'll be able to contribute to and closely follow a
"Guidelines to submitters" page in the future wiki-version of the
OEIS.

Benoit


2009/7/2  <zbi74583.boat at orange.zero.jp>:
>    Hagen, Benoit, Franklin, Richard, Robert.
>
>    Thank you for editing my sequence and telling me many opinions.
>    I feel the relationship between Odd perfect number conjecture and the
> sequence is interesting.
>
>    But don’t forget the original “PF sequence” which is written in the mail ti
> tled the same name. I posted it to the mailing list in April
>
>    A151659 is one of the easiest case of PF sequence, so the definition has
> several unnecessary descriptions.
>    I think Mathar’s definition is much better than mine.
>
>    >And I encourage Hagen to submit the equivalent sequence with sigma
> instead of usigma: to have a well-defined sequence, you can write "and
> a(n)=0 if the auxiliary sequence never reaches 1".
>
>
>    Wait !     Don’t leave me alone
>    See the mail. I already posted the case of Sigma and UnitaryPhi.and
> (-1)Sigma.
>    I am going to submit all these sequences.
>
>    I wonder if any better definition for the most generalized case exists.
>    I think the definition for PF sequence is rather good.
>
>
>
>    Yasutoshi
>
>    PS
>    Once I posted this mail but it didn't appear on the mailing list.
>    So, I post it again.
>
>
>
>        [seqfan] PF Sequence
> zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
> Sun Apr 5 08:18:33 CEST 2009
> • Previous message: [seqfan] Re: Dividing by a constant, policy?
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> ________________________________________
>    Hi, Seqfans
>    I found an intersting sequence related with divisor function.
>
>    [Definition of PrimeFactor_p or PF_p]
>
>         PrimeFactor_p[n] = The largest power of p diiding n
>         It is written as PF_p[n]
>
>
>         PF_p,q,r[n] = PF_p[n]* PF_q[n]* PF_r[n]
>         PF_1[n] = 1
>
>    [How to compute a(m)]
>    Case of Base Primes = {2}{3,5}
>
>    A=2^m, B=2^m
>    Do[
>         A=A/PF_2[UnitarySigma[B]]*PF_3,5[UnitarySigma[B]]
>         B=UnitarySigma[B]/ PF_2,3,5[UnitarySigma[B]]
>
>         IF B=1 THEN END   ]
>
>    a(m) = A
>
>    [Example]
>    m=5
>    A=2^5 , B=2^5
>    T=1
>         A=2^5/PF_2[3*11]* PF_3,5[3*11] =2^5*3
>         B=3*11/ PF_2,3,5[3*11] =11
>
>    T=2
>         A=2^5*3/PF_2[2^2*3]* PF_3,5[2^2*3] =2^3*3^2
>         B=2^2*3/ PF_2,3,5[2^2*3] = 1
>
>    a(5)=2^3`3^2
>
>         Base Prime={2}{3,5} UnitarySigma
>         b1(m) : 1, 2*3, 2^2*5, 2^3*3^2, 2^3*3^2, 2^3*3^2, 2^2*5, 2^3*3^2,
> 2^3*3^2, 2^7*3^3*5, 2^4*3*5^2, 2^7*3^3*5, 2^3*3^2
>
>         Base Prime={2}[3}
>         b2(m) : 1, 6, 6, 72, 72, 72, 6, 72, 72, 5184, 6, 5184, 72
>
>         Base Prime={2}{1}
>         b3(m) : 1, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
> 1/4
>
>         I don't understand well what the difference between m<14 2^m and 2^14
> is.
>
>    [How to compute a(m)  more generalized case]
>    F(n) means divisor function such as Sigma or UnitaryPhi or (-1)Sigma etc.
>
>    Case of Base Primes = {p}{q_i} , 1<=i<=k
>
>    A=p^m, B=p^m
>    Do[
>         A=A/PF_p[F [B]]*PF_{q_i , 1<=i<=k } [F [B]]
>         B=F [B]/ PF_{p, q_i , 1<=i<=k } [F [B]]
>
>         IF B=1 THEN END   ]
>
>    a(m) = A
>
>         Base Prime = {2}{1} UnitaryPhi
>         c3(m) : 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2
>
>         Base Prime = {2}{3}
>         c2(m) : 1, 2, 12, 12, 12, 12, 2^5*3^3, 2^5*3^3, 12, 2^5*3^3, 2^5*3^3
>
>         Base Prime = {2}{3,5}
>         c1(m) : 1, 2, 2^2*3, 2^2*3, 2^4*3*5, 2^4*3*5, 2^5*3^3, 2^5*3^3,
> 2^4*3*5, 2^5*3^3, 2^8*3^2*5^2
>
>         Base Prime = {2}{3,7}
>         c0(m) : 1, 2, 2^2*3, 2^2*3, 2^2*3, 2^2*3, 2^6*3^2*7, 2^6*3^2*7, 2^2*3,
> 2^6*3^2*7, 2^5*3^3
>
>         Base Prime = {2}{1} Sigma
>         d3(m) : 1, 1/2, 1/2, 1/4, 1/2, 1/4, 1/2, 1/8, 1/4, 1/2, 1/8
>
>         Base Prime = {2}{3} Sigma
>         d3(m) : 1, 2*3, 1/2, 2^2*3^2, 1/2, 2^2*3^2, 1/2, 2^5*3^4, 2^2*3^2,
> 2^2*3^2, 2^5*3^4
>
>         Base Prime = {3}{2} UnitarySigma
>         b4(m) : 1, 2^2*3, 2^2*3, 2^5*3^3, 2^5*3^3, 2^8*3^5, 2^7*3^4
>
>         Base Prime = {2}{1} (-1) Sigma
>         f3(m) : 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1
>
>         Base Prime = {2}{3} (-1) Sigma
>         f2(m) : 1, 2, 1,2*3, 2*3, 2*3, 2*3, 1, 2^4*3^3, 2*3, 2^4*3^3
>
>
>
>    Yasutoshi
>
>
>
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