A096290 Prime power perfect numbers: If n = Product p_i^r_i let PPsigma(n) = Product {Sum p_i^s_i, 2<=s_i<=r_i, s_i is prime}; sequence gives numbers k such that PPsigma(k) = 2*k.

216, 5400, 10584, 26136, 36504, 62424, 77976, 114264, 181656, 207576, 264600, 295704, 363096, 399384, 477144, 606744, 653400, 751896, 803736, 912600, 969624, 1088856, 1149984, 1151064, 1280664, 1348056, 1488024, 1560600, 1710936, 1788696, 1949400, 2032344, 2203416

Offset: 1

Author

Yasutoshi Kohmoto, Jun 24 2004

Keywords

• nonn

Examples

			5400 is in the sequence because 5400 = 2^3*3^3*5^2 and (2^2+2^3)*(3^2+3^3)*(5^2) = 2*5400.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A100509.

Programs

Maple

PPsigma := proc(n)
    option remember;
    local a, pe, p, e,f,i ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        p := op(1, pe) ;
        e := op(2, pe) ;
        f := 0 ;
        for i from 2 to e do
            if isprime(i) then
                f := f+p^i ;
            end if;
        end do:
        a := a*f ;
    end do;
    a ;
end proc:
for n from 1 do
    if PPsigma(n) = 2*n then
        print(n) ;
    end if;
end do: # R. J. Mathar, Mar 13 2024

Mathematica

f[p_, e_] := Plus @@ (p^Select[Range[e], PrimeQ]); s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300000], s[#] == 2*# &] (* Amiram Eldar, Sep 19 2022 *)

Extensions

Corrected and extended by Farideh Firoozbakht, Nov 17 2004

More terms from Amiram Eldar, Sep 19 2022