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%I #30 Feb 07 2020 03:07:47
%S 4704,1476096,396386304,6753750274277376,442715102395357986816,
%T 113336363243719574421504,31901471869127420013759771876790370304,
%U 42404329554681223873219247037048711787234652848116929825491652260298489856
%N Numbers of the form 2^(2*p-1)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.
%C Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}. The first element of the spectral basis of a(n) is A330819(n+1), the second element is A330837(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 The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 2*a(n)+1. In this case, we say that a(n) has index 2.
%C a(n) is also isospectral with A330838(n), that is, a(n) and A330838(n) have the same spectral basis, but A330838(n) has index 1. Thus, A330838(n) and a(n) form an isospectral pair.
%C 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 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 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) = A139306(n+1) * 3 * A133049(n+1).
%e If p = 3, then a(1) = 2^(2*3-1)*3*7^2 = 4704, and the spectral basis of 4704 is {63^2, 56^2, 48^2}, consisting of powers. The spectral sum of a(1), that is, the sum of the elements of its spectral basis, is 2*4704+1 = 9409. In this case, we say that a(1) has index 2. The number A330838(1) = 9704 has the same spectral basis as a(1), but with index 1. We say that A330838(1) and a(1) are isospectral and form an isospectral pair.
%p a := proc(n::posint)
%p local p, m;
%p p:=NumberTheory[IthMersenne](n+1);
%p m:=2^p-1;
%p return 2^(2*p-1)*3*m^2;
%p end:
%t f[p_] := 2^(2*p - 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* _Amiram Eldar_, Jan 12 2020 *)
%Y Cf. A000043, A000668, A133049, A139306, A330819, A330820, A330837, A330838.
%K nonn
%O 1,1
%A _Walter Kehowski_, Jan 12 2020
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