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.

28, 496, 33550336, 2658455991569831744654692615953842176

Offset: 1

Author

Jaroslav Krizek, Dec 04 2019

Keywords

• nonn

Comments

Corresponding values of even perfect numbers m: 6, 28, 8128, 2305843008139952128, ... (A330163).

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.

Crossrefs

Cf. A000043, A000396, A330163.

Programs

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]

Mathematica

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 *)