A330837 - OEIS

%I #32 Feb 10 2020 23:05:42

%S 144,3136,984064,264257536,4502500182851584,295143401596905324544,

%T 75557575495813049614336,21267647912751613342506514584526913536,

%U 28269553036454149248812831358032474524823101898744619883661101506865659904

%N a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

%C a(n+1) is the second element of the power-spectral basis of both A330836(n) and A330838(n). Also, a(n) = A139256(n)^2, where A139256(n) is the sum of the divisors of the n-th perfect number, A000396(n).

%C Also: squares of twice the perfect numbers. - _M. F. Hasler_, Feb 07 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) = A330824(n) * A133049(n).

%F a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - _M. F. Hasler_, Feb 07 2020

%e If p=3, then a(2) = (7*2^3)^2 = 56^2, and the spectral basis of A330836(1) = 4704 and A330838(1) = 9408 is {63^2, 56^2, 48^2}, consisting of powers.

%p a := proc(n::posint)

%p   local p, m;

%p   p:=NumberTheory[IthMersenne](n);

%p   m:=2^p-1;

%p   return m^2*(m+1)^2;

%p end:

%t f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* _Amiram Eldar_, Jan 12 2020 *)

%o (PARI) forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ _M. F. Hasler_, Feb 07 2020

%Y Cf. A000043, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jan 12 2020