PPS perfect number

[y.kohmoto]

    Hello, Farideh.
    Thanks you for the calculation of Prime power sigma perfect number.

    I mean that A096290 is "Primitive" Prime power sigma perfect number
    The non-primitive PPSigma-perfect numbers are obtained from the
primitive ones by multiplying by m, if m is the form : Product p_i^2  and
relatively prime to the primitive PPSigma-perfect number.
    Your example :
    5400=2^3*3^3*5^2 is represented as {Primitive PPSigma-perfect
number=2^3*3^3}*{p^2=5^2}

    These examples are not so interesting.
    Generally, Mathematicians don't count non-primitive solutions.

    But I think OEIS should have both sequences :
    %I A096290
    %S A096290 216,5400,10584,26136,36504,62424,77976,114264,181656,207576,
    %T A096290
264600,295704,363096,399384,477144,606744,653400,751896,803736
    %U A096290 912600,969624,1088856,1149984,1151064,1280664,1348056,1488024
    %N A096290 Prime power sigma perfect number.
    %C A096290 Farideh Firoozbakht caluculated most of terms.
    %Y A096290 A000001
    %K A096290 nonn
    %O A096290 1, 1
    %A A096290 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)


    %I A000001
    %S A000001 216, 1149984, 365727111552, 1590733122516339863159808000
    %N A000001 Primitive prime power sigma perfect number.
                    If n=Product p_i^r_i
                    then PPsigma(n)= Product {Sum p_i^s_i , 2<=s_i<=r_i ,
s_i is prime}
                    PPsigma(n)=2*n
    %e A000001 PPsigma(2^5*3^3)=(2^2+2^3+2^5)*(3^2+3^3)=1584
                     All powers of the terms are prime.
    %C A000001 Factorizations :
                    2^3*3^3, 2^5*3^3*11^3, 2^7*3^3*11^3*43^3,
2^13*3^5*5^3*19^3*23^3*31^3*137^3
                    No square free solution exists. All primes are cubic at
least.
                    The non-primitive PPSigma-perfect numbers are obtained
from the primitive ones by multiplying by m, if m is the form : Product
p_i^2  and relatively prime to the primitive PPSigma-perfect number.
    %Y A000001 A096290
    %K A000001 nonn
    %O A000001 1, 1
    %A A000001 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)


    A095682 : 1PPSigma perfect number has the same property.
    So, it should be rewritten.

    Yasutoshi