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%I
%S 6,264,45408,10177920,9310826880,27806077440,25437179036160,
%T 303753589954560,277875743791011840,14504815632384,13269098919960576,
%U 2534919599177957376,2318960803647990104064
%N Odd power perfect numbers: numbers k such that opsigma(k) = 2*k, where opsigma(k) = A033634(k).
%C If x is OPP and x=2^k*y, gcd(2^k,y)=1, (2^(k+4)+1)/3 is prime, then 4*x*(2^(k+4)+1)/3 is also OPP.
%C By applying the rule above to a(12) we get that 1772040615644549459607552 is also a term. - _Amiram Eldar_, Aug 26 2022
%F {k: A033634(k) = 2*k}.
%t f[e_] := If[OddQ[e], e+2, e+1]; fun[p_, e_] := 1 + (p^f[e] - p)/(p^2 - 1); opsigma[1] = 1; opsigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[50000], opsigma[#] == 2*# &] (* _Amiram Eldar_, Aug 26 2022 *)
%Y Cf. A000978, A033634.
%K nonn,more
%O 1,1
%A _Yasutoshi Kohmoto_
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