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Even perfect numbers m from A000396 such that w = (m + 2^(k(m) - 1) - 1) * 2^(2*(k(m) - 1)) is also an even perfect number, where k(m) is the Mersenne exponent A000043(m).
1

%I #15 Sep 08 2022 08:46:24

%S 6,28,8128,2305843008139952128

%N Even perfect numbers m from A000396 such that w = (m + 2^(k(m) - 1) - 1) * 2^(2*(k(m) - 1)) is also an even perfect number, where k(m) is the Mersenne exponent A000043(m).

%C Corresponding values of even perfect numbers w: 28, 496, 33550336, 2658455991569831744654692615953842176, ... (A330164).

%C Corresponding values of Mersenne exponents k(m) and k(w): (2, 3, 7, 31, ...), (3, 5, 13, 61, ...), where k(w) = 2*k(m) - 1.

%t f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; g /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* _Amiram Eldar_, Dec 06 2019 *)

%o (Magma) [(2^k - 1) * (2^(k - 1)): k in [1..100] | SumOfDivisors((2^k - 1) * (2^(k - 1))) / ( (2^k - 1) * (2^(k - 1))) eq 2 and SumOfDivisors(((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) / (((2^k - 1) * (2^(k - 1)) + (2^(k - 1) - 1)) * (2^(2*(k - 1)))) eq 2]

%Y Cf. A000043, A000396, A330164.

%K nonn,more

%O 1,1

%A _Jaroslav Krizek_, Dec 04 2019