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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

Table of n, a(n) for n=1..9.

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.)