SENSigma

[koh]

    Hi, Seqfans

    I submitted four sequences related with SENSigma and SEPSigma.
    
    A000001 - A000004 are numbered A125139 - A125142.


    I am sure that A000004 is a finite sequence.
    I mean that a number m exists such that {k -> infinity} -> {SEPSigma^{k}(m) -> infinity}.
    So k doesn't exist for m.
    Hence the sequence ends.

    I would like someone to search the smallest such m.

    I also want to know the following constants for several i.

    SENSigmaZeta(i)=Sum_k SENSigma(k)/k^i=1/1^i+3/2^i+4/3^i+....
    SEPSigmaZeta(i)=Sum_k SEPSigma(k)/k^i=1/1^i+1/2^i+2/3^i+....

 
    %I A000001
    %S A000001 1,3,4,5,6,12,8,15,11,18,12,20,14,24,24,29,18,33,20,30
    %N A000001 SENSigma(n) 
               SENSiguma(m) = (-1)^((Sum_i  r_i)+Omega(m))*Sum_{d|m} (-1)^((Sum_j r_j)+Omega(d))*d
                 =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1)
    
               Where m=Product_i p_i^r_i , d=Product_j p_j^r_j
               SEN for Signed by Exponents of prime factors and Number of prime factors.
    %e A000001 SENSigma(240)=(-1+2+4+8+16)*(1+3)*(1+5)
    %F A000001 SENSigma(n)=(p_i^(r_i+1)-p_i)/(p_i-1)+(-1)^(r_i+1)
    
    %Y A000001 A000002, A000003, A000004
    %K A000001 none
    %O A000001 1,2
    %A A000001 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp



    %I A000002
    %S A000002 1,1,2,7,4,2,6,13,13,4,10,14,12,6,8,31,16,13,18,28
    %N A000002 SEPSigma(n).
               SEPSigma(m) = (-1)^(Sum_i  r_i)*Sum_{d|m} (-1)^(Sum_j r_j)*d
                           =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^r_i

                           Where m=Product_i p_i^r_i , d=Product_j p_j^r_j
               SEP for Signed by Exponents of Prime factors .
    %e A000002 SEPSigma(240)=(1+2+4+8+16)*(-1+3)*(-1+5)
    %F A000002 SEPSigma(n)=(p_i^(r_i+1)-p_i)/(p_i-1)+(-1)^r_i
    %Y A000002 A000001, A000003, A000004
    %K A000002 none
    %O A000002 1,3
    %A A000002 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp



    %I A000003
    %S A000003 2,3,4,5,6,12,20,30,72,165
    %N A000003 a(n)=SENSigma(a(n-1)).
               SENSiguma(m) = (-1)^((Sum_i  r_i)+Omega(m))*Sum_{d|m} (-1)^((Sum_j r_j)+Omega(d))*d
                            =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^(r_i+1)
    
                            Where m=Product_i p_i^r_i , d=Product_j p_j^r_j
        
    %Y A000003 A000001, A000002, A000004
    %K A000003 none
    %O A000003 2,1
    %A A000003 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp
 


    %I A000004
    %S A000004 0,1,2,4,5,2,3,6,6,5
    %N A000004 The smallest k for SEPSigma^{k}(n)=1.
               SEPSigma(m) = (-1)^(Sum_i  r_i)*Sum_{d|m} (-1)^(Sum_j r_j)*d
                           =Product_i (Sum_{1<=s_i<=r_i} p_i^s_i)+(-1)^r_i

               Where m=Product_i p_i^r_i , d=Product_j p_j^r_j
    %e A000004 SEPSigma^{5}(5)=1, so a(5)=5.
               5 -> 4 -> 7 -> 6 -> 2 -> 1

    %Y A000004 A000001, A000002, A000003
    %K A000004 none
    %O A000004 1,3
    %A A000004 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp

    Yasutoshi
    
    



Dear SeqFans and Assoc. Eds.:
A few items:

- There is a new file "names.gz" (see the Welcome page)

- I was away for a week. I've just about caught up

- The "listen" button isn't working properly, due to

- "Find Friends" is working again, thanks to Russ Cox

- Many new entries have the "uned" keyword. If you want to

- The ban on junk submissions will continue indefinitely.

- Finally, if you are sending me coments in the middle of an email message,

%I A000010
%S A000010 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,
%T A000010 12,28,8,30,16,20,16,24,12,36,18,24,16,40,12,42,20,24,22,46,16,
%U A000010 42,20,32,24,52,18,40,24,36,28,58,16,60,30,36,32,48,20,66,32,44
%N A000010 Euler totient function phi(n): count numbers <= n and prime to n.
etc etc
- for then my editing program does a lot of the updating

But if you just say
%S A000010 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,
%T A000010 12,28,8,30,16,20,16,24,12,36,18,24,16,40,12,42,20,24,22,46,16,
%U A000010 42,20,32,24,52,18,40,24,36,28,58,16,60,30,36,32,48,20,66,32,44
%N A000010 Euler totient function phi(n): count numbers <= n and prime to n.
etc etc

with no %I line then I have to a lot more cutting and mousing,
which is always risky.

Regards
Neil