%I #9 May 14 2019 11:02:33

%S 36,392,30752,2064512,549621604352,2251765454077952,

%T 144114638320566272,9903520305059670164485701632,

%U 12259964326927110856232952250923146488025299504439754752,237142198758023568227473376531545064850552499416058362196624559881526665860349952

%N Primitive coreful perfect numbers: powerful numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

%C All the coreful perfect numbers (A307958) can be obtained from a primitive term k by multiplying it by m, if m is squarefree and coprime to k. The primitive terms are powerful. If k = m * r is in the sequence where r = rad(k) is the squarefree kernel of k (A007947), then the sum of the coreful divisors of k is csigma(k) = csigma(m * r) = sigma(m) * r, where sigma(m) is the sum of all the divisors of m (A000203). Thus k is a primitive coreful perfect number, iff sigma(m) * r = 2 * m * r, or m = k/rad(k) is a perfect number. Since k is powerful k = m * rad(m), thus all the primitive coreful perfect numbers can be generated from the perfect numbers by multiplying them by their squarefree kernel. The even terms of these sequence are (2^p) * (2^p-1)^2, were p is a Mersenne exponent (A000043). There is an odd term in this sequence iff there is an odd perfect number.

%F a(n) = A064549(A000396(n)) = A000396(n) * A007947(A000396(n)).

%e The first coreful perfect numbers are 36, 180 = 36*5, 252 = 36*7, 392, 396 = 36*11, 468 = 36*13, ... thus the primitive ones are 36, 392, ...

%t f[p_] := 2^p*(2^p-1)^2; f/@MersennePrimeExponent/@Range[10] (* assuming no odd perfect number exists *)

%Y Cf. A000043, A000203, A000396, A000668, A007947, A057723, A064549, A307958.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 08 2019