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A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

%I A330837 #32 Feb 10 2020 23:05:42
%S A330837 144,3136,984064,264257536,4502500182851584,295143401596905324544,
%T A330837 75557575495813049614336,21267647912751613342506514584526913536,
%U A330837 28269553036454149248812831358032474524823101898744619883661101506865659904
%N A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.
%C A330837 a(n+1) is the second element of the power-spectral basis of both A330836(n) and A330838(n). Also, a(n) = A139256(n)^2, where A139256(n) is the sum of the divisors of the n-th perfect number, A000396(n).
%C A330837 Also: squares of twice the perfect numbers. - _M. F. Hasler_, Feb 07 2020
%H A330837 G. Sobczyk, <a href="https://garretstar.com/secciones/publications/docs/monthly336-346.pdf">The Missing Spectral Basis in Algebra and Number Theory</a>, The American Mathematical Monthly 108(4), April 2001.
%H A330837 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent_(ring_theory)">Idempotent (ring theory)</a>
%H A330837 Wikipedia, <a href="https://en.wikipedia.org/wiki/Peirce_decomposition">Peirce decomposition</a>
%F A330837 a(n) = A330824(n) * A133049(n).
%F A330837 a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - _M. F. Hasler_, Feb 07 2020
%e A330837 If p=3, then a(2) = (7*2^3)^2 = 56^2, and the spectral basis of A330836(1) = 4704 and A330838(1) = 9408 is {63^2, 56^2, 48^2}, consisting of powers.
%p A330837 a := proc(n::posint)
%p A330837   local p, m;
%p A330837   p:=NumberTheory[IthMersenne](n);
%p A330837   m:=2^p-1;
%p A330837   return m^2*(m+1)^2;
%p A330837 end:
%t A330837 f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* _Amiram Eldar_, Jan 12 2020 *)
%o A330837 (PARI) forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ _M. F. Hasler_, Feb 07 2020
%Y A330837 Cf. A000043, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836.
%K A330837 nonn
%O A330837 1,1
%A A330837 _Walter Kehowski_, Jan 12 2020