Divisor

[koh]

    Hi, Seqfans


    I defined four divisor functions which are "difference" of divisors.

    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
    

    ex.    SENSigma(240)=(-1+2+4+8+16)*(1+3)*(1+5)
           SEN for Signed by Exponents of prime factors and Number of prime factors.

    
    2.
   
    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


    ex.    SEPSigma(240)=(1+2+4+8+16)*(-1+3)*(-1+5)
           SEP for Signed by Exponents of Prime factors .
      
    
    3.

    SENUnitarySigma 


    4.

    SEPUnitarySigma




    [SENSigma Multiply Perfect Number]

    SENSigma(m)=k*m , for some integer k.

    2^11*3^3*5^4*7^3*13*19*41     k=5
    2^9*3*11*31                   k=3
    2^9*3^3*5*11*31               k=4
    2^9*3^3*5^2*11*29*31          k=4
    2^9*3^4*7*11*17*31            k=4
    2^9*3*11^2*31*131             k=3
    2^9*3^3*5*11^2*31*131         k=4
    2^9*3^3*5^2*11^2*29*31*131    k=4
    2^9*3^4*7*11^2*17*31*131      k=4
    2^6*3*5^4*7*19*41             k=4
    2^6*3^2*5^4*7*11*19*41        k=4
    2^6*3^4*5^4*7^2*11*17*19*41   k=5  
    2^6*3^2*5^4*7*11^2*19*41*131  k=4
    2^6*3^4*5^4*7^2*11^2*17*19*41*131      k=5  
    2^5*3*7                                k=3
    2^5*3^2*7*11                           k=3 
    2^5*3^3*5*7                            k=4
    2^5*3^3*5^2*7*29                       k=4
    2^4*3*5*29                             k=3
    2^4*3^2*5*11*29                        k=3
    2^4*3^2*5*11^2*29*131                  k=3
    2^4*3^4*5*7*17*29                      k=4
    2^3*3*5*29                             k=3
    2^3*3^2*5*11*29                        k=3
    2^3*3^2*5*11^2*29*131                  k=3
    2^3*3^4*5*7*17*29                      k=4
    2^2*3*5                                 k=2
    2^2*3*5^2*29                            k=2
    2^2*3^2*5*11                            k=2
    2^2*3^2*5^2*11*29                       k=2
    2^2*3^2*5*11^2*131                       k=2
    2^2*3^2*5^2*11^2*29*131                  k=2
    2*3                                       k=2
    2*3^2*11                                  k=2
    2*3^2*11^2*131                            k=2



    [SEPSigma Multiply Perfect Number]

    SEPSigma(m)=k*m , for some integer k.


    2^10*3^5*5*7*11^2*19*23*89       k=2
    2^9*3*5*17*1021                  k=1
    2^8*3^5*5*7*73*181               k=2
    2^8*3^6*7^2*13*19*1093           k=3
    2^7*3^5*5*7*11^2*19*23*181       k=2
    2^6*3^5*5*7*127*181              k=2
    2^6*3^6*7^2*13*19*127*1093       k=3
    2^5*3*5*61                       k=1
    2^4*3*5*31                       k=1
    2^3*3*13                          k=1
    2^2*3*7                           k=1
    2*3                                k=1



    Is the following conjecture correct?

         "All SEPSigma Perfect Numbers are of the form 2^n*3^5*5*181*k, GCD(2*3*5*181,k)=1"


    Yasutoshi