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A330838 Numbers of the form 2^(2*p)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

%I A330838 #21 Feb 23 2020 15:22:16
%S A330838 9408,2952192,792772608,13507500548554752,885430204790715973632,
%T A330838 226672726487439148843008,63802943738254840027519543753580740608,
%U A330838 84808659109362447746438494074097423574469305696233859650983304520596979712
%N A330838 Numbers of the form 2^(2*p)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.
%C A330838 a(n) has the same spectral basis as A330836(n), namely {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}, so the two numbers are isospectral as well as power-spectral, that is, they have the same spectral basis and that basis consists of powers. The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 1*a(n)+1, while the spectral sum of A330836(n) is 2*A330836(n)+1. We say that a(n) and A330836(n) form an isospectral pair, with a(n) of index 1 and A330836(n) of index 2.
%C A330838 Subsequence of Zumkeller numbers (A083207), since a(n) = 2^r * 3 * s, where s is relatively prime to 6. - _Ivan N. Ianakiev_, Feb 03 2020
%H A330838 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 A330838 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent_(ring_theory)">Idempotent (ring theory)</a>
%H A330838 Wikipedia, <a href="https://en.wikipedia.org/wiki/Peirce_decomposition">Peirce decomposition</a>
%F A330838 a(n) = A330824(n+1) * 3 * A133049(n+1).
%e A330838 If p = 3, then M_3 = 7 and a(1) = 2^(2*3)*3*7^2 = 9408, with spectral basis {63^2, 56^2, 48^2}, and spectral sum equal to 1*9408 + 1 = 9409. However, {63^2, 56^2, 48^2} is also the spectral basis of A330836(1) = 4704, with spectral sum equal to 2*4704+1.
%p A330838 a := proc(n::posint)
%p A330838   local p, m;
%p A330838   p:=NumberTheory[IthMersenne](n+1);
%p A330838   m:=2^p-1;
%p A330838   return 2^(2*p)*3*m^2;
%p A330838 end:
%t A330838 f[p_] := 2^(2p)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9]  (* _Amiram Eldar_, Jan 17 2020 *)
%Y A330838 Cf. A000043, A000668, A133049, A330819, A330820, A330824, A330825, A330826, A330836, A330837.
%K A330838 nonn
%O A330838 1,1
%A A330838 _Walter Kehowski_, Jan 17 2020