A330838 Numbers of the form 2^(2*p)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

9408, 2952192, 792772608, 13507500548554752, 885430204790715973632, 226672726487439148843008, 63802943738254840027519543753580740608, 84808659109362447746438494074097423574469305696233859650983304520596979712

Offset

1,1

Comments

a(n) has the same spectral basis as A330836(n), namely {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}, so the two numbers are isospectral as well as power-spectral, that is, they have the same spectral basis and that basis consists of powers. The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 1*a(n)+1, while the spectral sum of A330836(n) is 2*A330836(n)+1. We say that a(n) and A330836(n) form an isospectral pair, with a(n) of index 1 and A330836(n) of index 2.

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

Links

Table of n, a(n) for n=1..8.

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+1) * 3 * A133049(n+1).

Example

If p = 3, then M_3 = 7 and a(1) = 2^(2*3)*3*7^2 = 9408, with spectral basis {63^2, 56^2, 48^2}, and spectral sum equal to 1*9408 + 1 = 9409. However, {63^2, 56^2, 48^2} is also the spectral basis of A330836(1) = 4704, with spectral sum equal to 2*4704+1.

Maple

a := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n+1);
  m:=2^p-1;
  return 2^(2*p)*3*m^2;
end:

Mathematica

f[p_] := 2^(2p)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9]  (* Amiram Eldar, Jan 17 2020 *)

Crossrefs

Cf. A000043, A000668, A133049, A330819, A330820, A330824, A330825, A330826, A330836, A330837.

Sequence in context: A183733 A290424 A210082 * A064587 A275922 A173194

Adjacent sequences: A330835 A330836 A330837 * A330839 A330840 A330841

Keyword

nonn

Author

Walter Kehowski, Jan 17 2020

Status

approved