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%I #17 Feb 02 2020 09:26:18
%S 576,12544,3936256,1057030144,18010000731406336,
%T 1180573606387621298176,302230301983252198457344,
%U 85070591651006453370026058338107654144,113078212145816596995251325432129898099292407594978479534644406027462639616
%N a(n) = 4*M(n)^2*(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.
%C Also a(n+1) is the second element of the power-spectral basis of A330839(n), where by power-spectral we mean that the spectral basis consists of primes and powers.
%H Garret 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, Vol. 108, No. 4 (April 2001), pp. 336-346.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent_(ring_theory)">Idempotent (ring theory)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Peirce_decomposition">Peirce decomposition</a>
%F a(n) = 4 * A133049(n) * A330824(n).
%e a(2) = 4*7^2*2^(2*3) = 2^8*7^2 = 112^2, and the spectral basis of A330839(1) = 18816 is {63^2, 112^2, 48^2}, consisting only of powers.
%p A330840 := proc(n::posint)
%p local p, m;
%p p:=NumberTheory[IthMersenne](n);
%p m:=2^p-1;
%p return 4*m^2*(m+1)^2;
%p end:
%t f[p_] := 2^(2*p + 2)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[9] (* _Amiram Eldar_, Jan 24 2020 *)
%Y Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330839.
%K nonn
%O 1,1
%A _Walter Kehowski_, Jan 23 2020
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