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

1,1

Links

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

Example

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.

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 *)

Crossrefs

Cf. A100509.

Sequence in context: A183617 A114060 A128964 * A246896 A017595 A232835

Adjacent sequences: A096287 A096288 A096289 * A096291 A096292 A096293

Keyword

nonn

Author

Yasutoshi Kohmoto, Jun 24 2004

Extensions

Corrected and extended by Farideh Firoozbakht, Nov 17 2004

More terms from Amiram Eldar, Sep 19 2022

Status

approved