[seqfan] PF Sequence

zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
Sun Apr 5 08:18:33 CEST 2009

```    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^-*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* ^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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