A330839 - OEIS

%I #19 Feb 23 2020 15:22:25

%S 18816,5904384,1585545216,27015001097109504,1770860409581431947264,

%T 453345452974878297686016,127605887476509680055039087507161481216,

%U 169617318218724895492876988148194847148938611392467719301966609041193959424

%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, 4*M_p^2*(M_p+1)^2, (M_p^2-1)^2}, where by power-spectral basis we mean a spectral basis that consists of primes and powers. The first element of the power-spectral basis is A330819(n+1), the second element is A330840(n+1), and the third element is A330820(n+1).

%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) = A330818(n+1) * 3 * A133049(n+1).

%e a(1) = 2^(2*3+1) * 3 * 7^2 = 18816, and 18816 has spectral basis {63^2, 112^2, 48^2}, consisting of powers.

%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^(2p + 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* _Amiram Eldar_, Jan 22 2020 *)

%Y Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330840.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jan 21 2020