comments on A001835, A004253, A001653

[Tanya Khovanova]

    Hi, Seqfans    

    Let S,T be subsets of set U={d|m} which satisfy the following conditions.

    o S cup T = U 
    o S cap T = Phi 
    o Sum_{d_i E S} d_i - Sum_{d_j E T} d_j = m 

    How many such partitions are there for each m?

    Example :
    m=12, S={1,3,4,12}, T={2,6}
          S={2,6,12}    T={1,3,4}
                   
    So, a(12)=2

    %S A000001 1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2

    %C A000001 If m=p^e then a(m)=0
               If m=Fixed points of (-1)Sigma and SEPSigma then 2<=a(m)




    (-1)Sigma(m) = (-1)^Omega(m)*Sum_{d|m} (-1)^Omega(d)*d

    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

    Yasutoshi