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A330841 Numbers of the form 2^(2*p-3)*9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

%I A330841 #27 Feb 10 2020 12:30:53
%S A330841 3528,1107072,297289728,5065312705708032,332036326796518490112,
%T A330841 85002272432789680816128,23926103901845565010319828907592777728,
%U A330841 31803247166010917904914435277786533840425989636087697369118739195223867392
%N A330841 Numbers of the form 2^(2*p-3)*9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.
%C A330841 a(1) = 3528 has power-spectral basis {21^2, 28^2, 48^2}, of index 1. If n > 1, then a(n) has power-spectral basis {M^2*(M+2)^2, (1/4)*M^2*(M+1)^2, (M^2-1)^2}, with index 2, where M=A000668(n+1) is the (n+1)-st Mersenne prime. The first element of the spectral basis of a(n), n > 1, is A330819(n+1), the second element is A133051(n+1), and the third element is A330820(n+1). Generally, a power-spectral basis is a spectral basis that consists of primes and powers.
%C A330841 The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is a(1) + 1 whenever n = 1, and 2*a(n)+1 whenever n > 1. In this case, we say that a(n) has index 1 and index 2, respectively.
%C A330841 a(n), n > 1, is also isospectral with 9*A133051(n), that is, a(n) and 9*A133051(n) have the same spectral basis, but 9*A133051(n) has index 1. Thus 9*A133051(n) and a(n) form an isospectral pair.
%H A330841 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 A330841 Wikipedia, <a href="https://en.wikipedia.org/wiki/Idempotent_(ring_theory)">Idempotent (ring theory)</a>
%H A330841 Wikipedia, <a href="https://en.wikipedia.org/wiki/Peirce_decomposition">Peirce decomposition</a>
%F A330841 a(n) = A152922(n+1) * 9 * A133049(n+1).
%e A330841 a(2) = 2^(2*5-3)*9*31^2 = 2^7*9*31^2 = 1107072 has spectral basis {1023^2, 496^2, 960^2}, consisting of powers. The spectral sum of a(2), that is, the sum of the elements of its spectral basis, is 2*a(2)+1 = 2214145. In this case we say that a(2) has index 2. The number 9 * A330817(2) = 2^(2*5-2)*9*31^2 = 2^8*9*31^2 = 2214144 has the same spectral basis as a(2), but with index 1. We say that 9 * A330817(2) and a(2) are isospectral and form an isospectral pair.
%p A330841 a := proc(n::posint)
%p A330841 local p, m;
%p A330841 p:=NumberTheory[IthMersenne](n+1);
%p A330841 m:=2^p-1;
%p A330841 return 2^(2*p-3)*9*m^2;
%p A330841 end;
%t A330841 f[p_] := 9*2^(2*p - 3)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* _Amiram Eldar_, Feb 07 2020 *)
%Y A330841 Cf. A000043, A000668, A133049, A133051, A152921, A152922, A330818, A330819, A330820, 9*A133051, 9*A330817, A330837.
%K A330841 nonn
%O A330841 1,1
%A A330841 _Walter Kehowski_, Jan 25 2020