[seqfan] PF seqence and A151659
zbi74583.boat at orange.zero.jp
zbi74583.boat at orange.zero.jp
Thu Jul 2 09:32:30 CEST 2009
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
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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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