login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 3
9408, 2952192, 792772608, 13507500548554752, 885430204790715973632, 226672726487439148843008, 63802943738254840027519543753580740608, 84808659109362447746438494074097423574469305696233859650983304520596979712 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 22:49 EST 2021. Contains 349596 sequences. (Running on oeis4.)