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 A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime. 3
 144, 3136, 984064, 264257536, 4502500182851584, 295143401596905324544, 75557575495813049614336, 21267647912751613342506514584526913536, 28269553036454149248812831358032474524823101898744619883661101506865659904 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). Also: squares of twice the perfect numbers. - M. F. Hasler, Feb 07 2020 LINKS G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001. Wikipedia, Idempotent (ring theory) Wikipedia, Peirce decomposition FORMULA a(n) = A330824(n) * A133049(n). a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - M. F. Hasler, Feb 07 2020 EXAMPLE 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. MAPLE a := proc(n::posint)   local p, m;   p:=NumberTheory[IthMersenne](n);   m:=2^p-1;   return m^2*(m+1)^2; end: MATHEMATICA f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *) PROG (PARI) forprime(p=1, 999, isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)", ")) \\ M. F. Hasler, Feb 07 2020 CROSSREFS Cf. A000043, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836. Sequence in context: A223323 A251433 A223300 * A252182 A187166 A231836 Adjacent sequences:  A330834 A330835 A330836 * A330838 A330839 A330840 KEYWORD nonn AUTHOR Walter Kehowski, Jan 12 2020 STATUS approved

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Last modified January 20 12:55 EST 2022. Contains 350472 sequences. (Running on oeis4.)