cp's OEIS Frontend

A330164 Even perfect numbers w from A000396 such that number m = w / 2^(k(w) - 1) - 2^((k(w) - 1)/2) + 1 = 2^k(w) - 2^((k(w) - 1)/2) is also an even perfect number, where k(w) is the Mersenne exponent (A000043) for number w.

%I A330164 #10 Sep 08 2022 08:46:24
%S A330164 28,496,33550336,2658455991569831744654692615953842176
%N A330164 Even perfect numbers w from A000396 such that number m = w / 2^(k(w) - 1) - 2^((k(w) - 1)/2) + 1 = 2^k(w) - 2^((k(w) - 1)/2) is also an even perfect number, where k(w) is the Mersenne exponent (A000043) for number w.
%C A330164 Corresponding values of even perfect numbers m: 6, 28, 8128, 2305843008139952128, ... (A330163).
%C A330164 Corresponding values of Mersenne exponents k(w) and k(m): (3, 5, 13, 61, ...), (2, 3, 7, 31, ...), where k(m) = (k(w) + 1)/2.
%t A330164 f[n_] := 2^(n - 1)*(2^n - 1); g[n_] := 2^n - 2^((n - 1)/2); mers = MersennePrimeExponent[Range[10]]; f /@ Select[mers, MemberQ[f /@ mers, g[#]] &] (* _Amiram Eldar_, Dec 06 2019 *)
%o A330164 (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 - 2^((k-1) div 2)) / (2^k - 2^((k-1) div 2) ) eq 2]
%Y A330164 Cf. A000043, A000396, A330163.
%K A330164 nonn
%O A330164 1,1
%A A330164 _Jaroslav Krizek_, Dec 04 2019