2^n-3

[y.kohmoto]

>
>Dear Yasutoshi,
>
>Please explain that to me what are the definitions of -Sigma(m)
>and -Sigma perfect number?
>
>Regards,
>
>Farideh
>
>
>
>Quoting "y.kohmoto" <zbi74583 at boat.zero.ad.jp>:
>
>>     Hello, seqfans.
>>     When I calculated -1Sigma perfect number, I factorized many times the
>> numbers of form 2^n-3.
>>     Because If m=Product p_i^r_i then -1Sigma(m)=Product(-1+Sum p_k^r_k ,
>> k=1 to i), so if p_i=2 then -1Sigma(2^n)=2^(n+1)-1-2=2^(n+1)-3.
>>     And it is necessary to factorize -1Sigma(2^n) for calculating -1Sigma
>> perfect number.
>>
>>     s1 : 1, 5, 13, 29, 61, 5, 11, 509, 1021, 5, 4093, 19, 16381, 5, 13
>>     s2 : 1, 5, 13, 29, 61, 5, 23, 509, 1021, 409, 4093, 431, 16381, 6553,
71
>>     s3 : 3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29,
>>     s4 : 5, 13, 29, 61, 509, 1021, 4093, 16381, 1048573, 4194301,
16777213,
>> 536870909,
>>
>>     s1 : the smallest prime factor of 2^n-3
>>     s2 : the greatest prime factor of 2^n-3
>>     s3 : numbers n such that 2^n-3 is prime
>>     s4 : primes of form 2^n-3
>>
>>     s3 already exists on OEIS.
>>
>>     To Neil :
>>     Do you need any of them?
>>
>>     Yasutoshi
>>
>>
>>
>


    Hello, Farideh

    -1Sigma(n) is one of the generalization of Sigma(n).
    It is defined as follows.

    If m=Product p_i^r_i then -1Sigma(m)=Product(-1+Sum p_i^r_k , r_k=1 to
r_i)

    Sorry, I realized that I had done a typo mistake.
    The definition in a first mail was -1Sigma(m)=Product(-1+Sum p_k^r_k ,
k=1 to i), it was false.

    The reason why it is a generalization of Sigma :
    Sigma(m) = Product(Sum p_i^r_k , r_k=0 to r_i)
                  = Product(1+p_i^1+p_i^2+....p_i^r_i)
    And,
    -1Sigma(m) = Product(-1+Sum p_i^r_k , r_k=1 to r_i)
                     = Product(-1+p_i^1+p_i^2+....p_i^r_i)
    The signature of the first term in a parentheses is minus,  so it
becomes a difference of divisors of m.
    example :
    -1Sigma(10000)=-1Sigma(2^4*5^4)=(-1+2+2^2+2^3+2^4)*(-1+5+5^2+5^3+5^4)=29
*779

    -1Sigma Perfect number is defined as follows.
    -1Sigma(n)=k*n , for some inteder k.

    More exactly it should be called a Multiple Perfect number.

    Please search the word "Kohmoto" and "Sigma" on the advance search site
in OEIS home page, then you will see many kind of generalization of Sigma
which I studied.
    They are also available on "Aliquot cycles and generalizations" in my
home page.
     http://boat.zero.ad.jp/~zbi74583/another02.htm
    One example :
    -1Sigma aliquot sequence of period 8.
    a(n)=-1Sigma(a(n-1))
    2^3*5*7*29 - 2^5*3*7*13 - 2^4*3^2*61 - 2^2*3*5*11*29 - 2^6*5^2*7 -
2*3*5^3*29 - 2^4*7^2*11 - 2*5^2*11*29

    Yasutoshi